Understanding Advanced Factoring. Factor Theorem For polynomial P(x), (x - a) is a factor of P(x) if and only if P(a) = 0.

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1 61 LESSON Understanding Advanced Factoring Warm Up New Concepts 1. Vocabulary A polynomial of degree one with two terms is a.. True or False: The GCF of -x 4 y 5 - xy + is xy. (11) (). Divide 4x - x + 5 by x - 1 using synthetic substitution. (51) Recall that synthetic division is used to divide a polynomial P(x) by a linear binomial of the form x - a and that the last number in the bottom row is the remainder and also the value of P(a). If the remainder is 0, the linear binomial is a factor of the polynomial. This is known as the Factor Theorem. Factor Theorem For polynomial P(x), (x - a) is a factor of P(x) if and only if P(a) = 0. Hint x + = x - (-), so the value of a is -. For example, x + is a factor of x + 10x + 16 and P(-) = (-) + 10(-) + 16 = = 0. Example 1 Determining Whether a Linear Binomial is a Factor Determine whether the linear binomial is a factor of the polynomial. a. P(x) = x - x - 4x + 60, x - 4 SOLUTION Use synthetic division with a = Because P(4) = 0, x - 4 is a factor of P(x) = x - x - 4x b. P(x) = x + x - 4x + 6, x + SOLUTION Use synthetic division with a = Because P(-) 0, x + is not a factor of P(x) = x + x - 4x + 6. Online Connection When the remainder is zero, the quotient of P(x) and (x - a) is another factor of the polynomial. This factor may be able to be further factored by other factoring methods. For instance, in Example 1a, the quotient x + 7x - 15 factors into (x - )(x + 5). Therefore, P(x) factors into (x - 4)(x - )(x + 5). 46 Saxon Algebra

2 If the remaining quotient is a cubic, it may be factorable by one of the following factoring methods. Sum and Difference of Cubes Sum of Two Cubes: a + b = (a + b)(a - ab + b ) Difference of Two Cubes: a - b = (a - b)(a + ab + b ) Math Reasoning Justify Tell why x - 1 is not a difference of cubes. Example Factoring the Sum and Difference of Two Cubes Factor each expression. a. d + 15 SOLUTION The expression is a sum of two cubes, where a = d and b = 5. d + 15 (d + 5)(d - 5d + 5) b. 7x 6 + y 9 SOLUTION The expression is a sum of two cubes, where a = x and b = y. 7x 6 + y 9 = (x ) + (y ) (x + y )(9x 4 - x y + y 6 ) c. m 5-4m SOLUTION First factor out the GCF, m. Then factor the difference of the two cubes, where a = m and b =. m 5-4m m (m - 8) m (m - )(m + m + 4) Another method of factoring is factoring by grouping. To factor by grouping, group the terms evenly so that each group has a common factor. Factor out the GCF from each group. Then factor out the common polynomial factor. Example Factor each expression. a. x + 5x + x + 15 Factoring by Grouping SOLUTION The first two terms have a common factor as do the last two terms. (x + 5x ) + (x + 15) x (x + 5) + (x + 5) Factor x from the first group and from the second. (x + 5)(x + ) Factor out the GCF of (x + 5). Neither factor can be factored further. Lesson 61 47

3 Math Reasoning Analyze Can a polynomial with five terms be factored by using the grouping method? Explain. b. x 4 + 4x - x - 4 SOLUTION Factor x from the first two terms and -1 from the last two terms. x 4 + 4x - x - 4 x (x + 4) - 1(x + 4) (x + 4)(x - 1) Factor out the GCF of (x + 4). (x + 4)(x - 1)(x + x + 1) Factor the difference of cubes. c. x 6 + x 5 + x 4 + x + x + SOLUTION Use two groups of three terms each. (x 6 + x 5 + x 4 ) + (x + x + ) x 4 (x + x + ) + 1(x + x + ) (x + x + )(x 4 + 1) Factor out the GCF of (x + x + ). (x + 1)(x + )(x 4 + 1) Factor the trinomial. Example 4 Application: Three-Dimensional Figures The volume of a rectangular solid with a length of x - 5 feet has a volume of x - x + 170x cubic feet. Factor the expression for the volume completely. SOLUTION Since volume is the product of the length, width, and height, the expression for the length, x - 5, is a factor of the polynomial. Use synthetic division to find the quotient of the polynomial and x The expression for the volume factors into (x - 5)(x - 18x + 80). But the trinomial can still be factored: (x - 10)(x - 8). The expression for the volume factors into (x - 5)(x - 10)(x - 8). Lesson Practice Determine whether the linear binomial is a factor of the polynomial. a. P(x) = x + 19x + x + 16, x + (Ex 1) (Ex 1) b. P(x) = 4x - x + 56x - 1, x - 6 Factor each expression. c h d. s 6 t 1-15r (Ex ) (Ex ) e. 8 7 x7 + x 4 f. xy - 9y + 5x - 45 (Ex ) (Ex ) (Ex ) g. 4m n + 1m - n - h. x 5 + 1x 4 + 6x + 5x + 60x (Ex ) (Ex ) (Ex 4) i. The volume of a rectangular solid with a length of x - 15 feet has a volume of x + 5x - 600x cubic feet. Factor the expression for the volume completely. 48 Saxon Algebra

4 Practice Distributed and Integrated Graph. 1. x = -5. x - y = 4. 7y + 7x = 1 (1) (1) (1) (Inv ) 4. Find the intercepts of the equation 5x - 4y + 7z = 1. Simplify (59) (59) (59) Find an inverse for the equations. 8. y = x - 9. y = 5x + 15 (50) (50) 10. Let K(u) = 17 and L(u) = -1. Find K (L(u)) and L (K(u)). (5) *11. Minimize P = 4x + y for the constraints (54) x 4 y 6 x + y 7 *1. Mary deposited $980 at 7.0% compounded continuously. How much money will (57) she have after 9 years? 1. Meteorology The number of tornadoes in the United States in 006 is shown in (55) the table. Season Jan-Feb Mar-Apr May-Jun Jul-Aug Sep-Oct Nov-Dec Number of Tornadoes What is the experimental probability that a tornado occurs in the first six months of the year? *14. Write Tell how factoring a sum of cubes is similar to factoring a difference of cubes. (61) 15. Probability A student graphed a set of data points and found that the correlation (45) coefficient was What is the probability that the slope of the line of best fit is greater than 1? Explain. *16. Geometry The probability that a randomly selected point lies within a (60) region can be found using the ratio of the area the point would lie in to the total area. Find the probability that a random point within the region lies within one of the smaller circles. 10 in. 6 in. in. Lesson 61 49

5 17. Buildings One building is 10 feet tall and a nearby building is 74 feet tall. From (46) the top of the taller building, you can spot the top of the shorter building at an angle of depression of 0. How far apart are the two buildings? Round to the nearest foot. 18. Walking Shadows While walking the dog one night, you notice how the length of (5) your shadow changes as you approach each lamppost. The length of the shadow seems to be 1 5 of your distance to the lamppost. Your dog is dragging you along at about feet per second, aiming for a lamppost. Show how to write a composite function that gives the length of your shadow in terms of time as you approach the lamppost. *19. Error Analysis Is the work below correct? If not, find and explain the error, and (59) give the correct solution. x 8 = ( x 8 ) 8 = () 8 x 1 = 8 x = Analyze Suppose you spin the spinner shown 8 times. What is the probability (49) that you will spin black 5 times? 1. Verify Use the form x + cx + c for a quadratic perfect square trinomial to verify (58) the method of completing the square. *. Graphing Calculator Describe the feasible region for the inequalities: (54) x 0, y 0, y -0.75x + 8. Multiple Choice Which value of r, the correlation coefficient, describes the set of (45) points that are closest to forming a line? A r = B r = 0 C r = 0.6 D r = 0.65 *4. Commuting to Work According to data collected in the 005 American Community (60) Survey, approximately 10.67% of workers carpool to work, 4.66% use public transportation, and.47% walk. Find the probability that a randomly selected worker carpools or uses public transportation. 5. Write A few different equations of the form y = b x are graphed to the (47) right. Describe in your own words how the shape of the graph depends on the value of b. y =.5 x O y y =.5 x y = 1.5 x x Saxon Algebra

6 *6. Multi-Step Use (59) 5 5 a. Which property of rational exponents should be applied first to simplify the expression? Explain. b. Apply the property from part a. c. Which property of rational exponents should be applied next? d. Apply the property from part c. e. If possible, finish simplifying the expression from part d. 7. Multiple Choice In the diagram to the right, find the length of the (5) hypotenuse of triangle LMN. A 7.5 B C 15 D 15 N L 15 cm M *8. Multi-Step The area of a twin size mattress is x x + 5x square inches. (61) a. Factor the polynomial by grouping. b. Find the area given that x = Amusement Parks The Giant Swing at Silver Dollar City in Branson, Missouri, rotates (56) through an angle measuring 0. Find the reference angle for this angle. *0. Verify Show that there are two different ways to group the terms in (61) d + 6f + d + df and that both lead to the same factorization. Lesson

7 6 LESSON Using Complex Numbers Warm Up New Concepts Math Reasoning Generalize For any positive real n, -n =. 1. Vocabulary Any point on the number line is a number. (1). Simplify (5 + x) - (6 - x). (). Solve 5x - 15 = 0. (58) 4. Simplify 150. (40) The equation x = ± -16 has no real solutions because there are no real numbers that, when squared, equal -16. But the equation does have solutions when a new set of numbers, outside of the real numbers, is introduced. These numbers are the imaginary numbers. Imaginary Numbers An imaginary number is written in the form bi where b is a real number and i is the imaginary unit. i is a solution of x = -1. Because i = -1, i = -1. The square root of a negative number is an imaginary number. -16 = 16-1 = 16-1 = 4i Example 1 Simplifying Square Roots of Negative Numbers a. Simplify SOLUTION Write -49 as the product of 49 and The square root of a product is the product of the square roots. - 7 i 49 = 7 and -1 = i. -14i Multiply the real numbers. b. Simplify Online Connection SOLUTION Write -117 as a product with a perfect square The square root of a product is the product of the square roots. 1 (i) 9 = and -1 = i. i 1 Commutative Property of Multiplication 44 Saxon Algebra

8 Example Solving a Quadratic Equation with Imaginary Numbers Solve x + 5 = 0. Write the solutions in terms of i. Check the answers. SOLUTION x + 5 = 0 x = -5 Subtract 5 from each side. x =± -5 Take the square root of each side. x =±5i Simplify the square root. Math Language The solutions are the conjugate pairs 5i and -5i. Complex solutions of quadratic equations always occur in conjugate pairs of the form a + bi and a - bi. Check x + 5 = 0 (5i) i (-1) = 0 x + 5 = 0 (-5i) i (-1) = 0 An imaginary number bi is part of a complex number. Math Reasoning Analyze Are real numbers a subset of complex numbers or are complex numbers a subset of real numbers? Complex Numbers A complex number is a number that can be written in the form a + bi, where a and b are real numbers. In a complex number, a is called the real part and bi is called the imaginary part. If b = 0, then the imaginary part is 0 and the number is a real number. If a = 0 and b 0, then the real part is 0 and the number is an imaginary number. Complex Real Imaginary + i + 0i = 0 + i = i To add or subtract complex numbers, add or subtract the real parts and then add or subtract the imaginary parts. Add or Subtract Complex Numbers (a + bi) + (c + di) = (a + c) + (b + d )i (a + bi) - (c + di) = (a - c) + (b - d )i Example Simplifying Expressions with Complex Numbers Add or subtract. Write the answer in the form a + bi. a. (7 + 4i) + (- - 5i) SOLUTION (7 + (-)) + (4 + (-5))i = 5 + (-1)i = 5 - i b. (5 - i) - (5-11i) SOLUTION (5-5) + (- - (-11))i = 0 + 8i = 8i Lesson 6 44

9 Example 4 Solving Quadratic Equations Solve each equation. Write the solutions in the form a + bi. a. (x - ) = -4 SOLUTION (x - ) = -4 x - = ± -4 Take the square root of each side. x - = ±i Simplify the square root. x - = i or x - = -i Solve both equations. x = + i or x = - i The solutions are + i and - i. b. x + 8x + 18 = 0 SOLUTION x + 8x = -18 x + 8x + 16 = Subtract 18 from each side. Complete the square. (x + 4) = - Write the left side as a perfect square. x + 4 = ± - Take the square root of each side. x + 4 = ±i Simplify the square root. x = -4 ± i Subtract 4 from each side. The solutions are -4 + i and -4 - i. Lesson Practice a. Simplify (Ex 1) b. Simplify (Ex 1) (Ex ) (Ex ) c. Solve 0 = x. Write the solutions in terms of i. Check the answer. d. Solve -6x = 16. Write the solutions in terms of i. Add or subtract. Write the answer in the form a + bi. (Ex ) e. (- + i) + (14 + 4i) f. (10-6i) - (-4 + i) Solve each equation. Write the solutions in the form a + bi. (Ex 4) g. (x + 9) = -9 h. x - x + = 0 Practice Distributed and Integrated Factor. *1. 15x x - x - *. 64x 1-15y 9 (61) (61) 444 Saxon Algebra

10 Add or subtract.. a + a (7) b 4. ax m p - c + m (7) (7) (7) gh 5. - x + xh + g h gx Factor the greatest common factor. 6. 5x 7 y 5 m - 7x 5 m y + 14y 7 x 4 m 7. 6x ym 5 - x ym + 4xym () () Simplify (-7) (59) (59) 81 4 (59) (59) School It is estimated that 15% of students walk to school. What is the probability (49) that exactly out of 5 randomly selected students will walk to school? 1. Find an equation for the inverse of y = 1 5 x + 6. (50) 1. What are the odds in favor of rolling a on a number cube? (55) *14. Multiple Choice Which is a real number? (6) A 15i B i C 5 + 0i D i *15. Verify Show that the expression (7) x + 1 is equivalent to x + 1 x +. *16. Electrical Engineering A circuit has a current of (4 - i) amps, and a second circuit (6) has a current of (6-6i) amps. Find the sum of the currents. 17. Analyze Explain why the odds in favor of an event with a probability of 50% are 1 to 1. (55) 18. Projectile Motion The Leaning Tower of Pisa measures feet. If an object is (58) thrown into the air, with an initial velocity of 80 feet per second, from the top of the lower side, the height, h, of the object in feet after t seconds can be described by the equation h = -16t + 80t Find the time at which the object will reach a height of 00 feet. Round your answer to the tenths place. *19. A number cube is rolled once. Are the events a number greater than or a multiple (60) of mutually exclusive or inclusive? Explain why. *0. Transportation The approximate volume of a moving truck, in cubic feet, can be (61) given by x - 6x + 8x + 5x - 0x Factor this expression by grouping. Then find the volume when x = Multiple Choice Which of the following angles is not coterminal with 5º? (56) A 585º B -15º C 95º D -495º. Multi-Step The population of an insect colony is growing at a rate of 1.5% per week. (57) a. Write an expression for the function N(P) that gives the population one week after the population is P. b. What composite function gives the population three weeks after the population is P? Lesson 6 445

11 *. Graphing Calculator Factor x 5 - x 4 - x + 1. Then enter the original expression for (61) Y1 and the factored expression for Y on a graphing calculator. Access the Table function and study the table of values for the expressions. What is true? What does it mean? 4. Write Explain why radical expressions with even indices and positive radicands (59) have two real roots. Use 6 64 to demonstrate your explanation. *5. Geometry An expression for the volume of the rectangular (51) prism is V = x + 7x - 14x Find an expression for the missing dimension. x + 6. Analyze Suppose the graph of y = b x + k passes through the (57) points (, -) and (5, 0). Determine whether the equation models exponential growth or decay. x + 4? 7. Error Analysis A student made an error while studying the relation (5) between the two triangles shown. Explain the error. Find the missing angles. m Y = 90-0 = 60 m R = = 0 m X = 90 m Y = 60 m Z = 0 Y X 0 Z P 60 Q m P = 90 m Q = 60 m R = 0 The corresponding angles are congruent. So, the triangles are congruent. R 8. Probability A sales clerk has a weekly quota of 15 sales. The probability, p, that the (58) sales clerk will meet this quota after working d days, in a 5-day work week, can be approximated by p = -0.04d + 0.4d. How many days a week should the sales clerk work for the probability of making 15 sales to be 0.8? *9. Verify Show that 11i and -11i are solutions of x = -11. (6) 0. Cell Phones Mobile-U has two cell phone plans, the Metropolitan Plan and the (54) Continental Plan. The cost for each plan is shown in the table. A company is comparing the two plans. The company needs at least 1 cell phones, one for each sales representative. Each representative will use at least 500 minutes per month. What options does this company have? Continental Plan Metropolitan Plan Start-up Fee $50 $100 Free minutes Per minute rate above free 10 cents 7 cents minutes Monthly charge $0 # of cell phones $5 # of cell phones 446 Saxon Algebra

12 6 LESSON Understanding the Unit Circle and Radian Measures Warm Up 1. Vocabulary Half the diameter of a circle is the of the circle. (SB). If both legs of a right triangle have length 1, then the length of the hypotenuse is. (41). If the shortest leg of a triangle has length 1, then the lengths of the other two sides of the triangle are and. (5) New Concepts A unit circle is a circle with a radius of 1 unit. Exploration Exploring the Unit Circle Use trigonometric ratios to determine the coordinates of points on a unit circle that is centered at the origin. 1. The figure shows a unit circle and a 60 angle in standard position. What is the length of the hypotenuse of the triangle? How do you know?. Use what you know about special right triangles to find the lengths of the legs of the triangle.. What are the coordinates of point P? 4. What are the exact values of cos 60 and sin 60? 5. How are the values of cos 60 and sin 60 related to the coordinates of point P? 6. Describe how you can use a similar method to find the coordinates of point Q. 7. Explain how the values of cos 45 and sin 45 are related to the coordinates of point Q y y x x 1 1 P(x, y) y y x Q(x, y) x Reading Math Remember the symbol θ is read theta. Online Connection Angles can be measured in degrees or in radians. The measure of an angle in radians is based on arc length, the distance between two points on a circle. In a circle with radius r, the measure of a central angle θ is one radian when it intercepts an arc that has a length equal to the radius. r θ = 1 radian r Lesson 6 447

13 There are 60 or π radians in a circle. Thus the conversion factor π 60, which is equivalent to 1, can be used to convert measures θ = -10 = - 7π 6 between degrees and radians. All angles are measured clockwise or counterclockwise from the positive x-axis. Measured clockwise, they are negative angles. In the figure, the -10 angle is equivalent to an angle measure of 150. y θ = 60 = π radians x Converting Angle Measures Degrees to Radians Radians to Degrees Multiply the number of degrees Multiply the number of radians by ( π radians 180 ). by 180 ( π radians ). Math Language Angle measures in radians often appear without the label radians. For example, - π 6 radian is often written simply as - π 6. Example 1 Converting Between Degrees and Radians Convert each measure from degrees to radians or from radians to degrees. a. -0 SOLUTION -0 ( π radians 180 ) = - π b. π 4 radians SOLUTION ( π 6 radians Multiply by ( π radians 180 ). 4 radians 180 ) ( π radians ) = 15 Multiply by ( 180 π radians ). For every point P(x, y) on the unit circle, the value of r is 1. Therefore, if θ is an angle in standard position whose terminal side passes through point (x, y) on the unit circle, then: sin θ = y r = y 1 = y cos θ = x r = x 1 = x tan θ = y x So, the coordinates of P can be written as (cos θ, sin θ). The Unit Circle ( 1 (0, 1) ( 1 π,, ) ) ( π π, ) 90 ( π π, ) ( π 5π ( 6, 1 ), 1 ) (-1, 0) π 180 II I 0 0 (1, 0) III IV π 11π π 5π 40 ( π 4, 1 (, 1 ) 4π π (, ( ) ), (0, -1) ( 1, ( 1 ), ) The diagram shows the equivalent degree and radian measures of special angles, and the corresponding x- and y-coordinates of points on the unit circle. ) 448 Saxon Algebra

14 Example Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find each trigonometric function value. Find exact values. a. sin 10 Hint The exact value of sin 10 is irrational. An approximate value is sin 10 = exact approximate SOLUTION The terminal side of 10 passes through the point (- 1, ) on the unit circle. sin θ = y sin 10 = b. tan 11π 6 SOLUTION The terminal side of 11π passes through the point 6 (, - 1 ) on the unit circle. tan θ = y x tan 1-11π 6 = = - 1 = - 1 = - 1 = - To divide by a fraction, multiply by its reciprocal. Rationalize the denominator. You can use reference angles and the portion of the unit circle in Quadrant I to determine trigonometric function values. Evaluate Trigonometric Functions using Reference Angles To find the sine, cosine, or tangent of θ: Step 1: Determine the measure of the reference angle of θ. Step : Use the portion of the unit circle in Quadrant I to find the sine, cosine, or tangent of the reference angle. Step : Use the quadrant of the terminal side of θ in standard position to determine the sign of the sine, cosine, or tangent. Lesson 6 449

15 The diagram shows how the signs of the trigonometric functions are determined by the quadrant that contains the terminal side of θ in standard position. QII QIII sinθ + sinθ + cosθ cosθ + tanθ tanθ + sinθ sinθ cosθ cosθ + tanθ + tanθ QI QIV Hint Recall that a reference angle is an acute angle. So for θ < 60, the measure of θ is: QII θ QIII - θ QIV θ Math Language An arc is an unbroken part of a circle consisting of two points on the circle, called endpoints, and all the points on the circle between them. This arc is named RS. R S Example Using Reference Angles to Evaluate Trigonometric Functions Find the sine, cosine, and tangent of 15. Find exact values. SOLUTION Step 1: Find the measure of the reference angle. The measure of the reference angle is 45. Step : Find the sine, cosine, and tangent of the reference angle. sin 45 = Use sin θ = y. cos 45 = Use cos θ = x. tan 45 = 1 Use tan θ = y x. Step : Determine the sign. sin 15 = In Quadrant II, sin θ is positive. cos 15 = - tan 15 = -1 In Quadrant II, cos θ is negative. In Quadrant II, tan θ is negative. The arc length of a circle intercepted by the central angle is related to the central angle. Arc Length Formula For a circle of radius r, the arc length s intercepted by a central angle θ (measured in radians) is given by the following formula. s = rθ 45 y 45 y 15 x x (, ) θ r s The arc length s of a circle is a portion of the circumference of a circle. This portion is the ratio between the measure of the central angle and the measure of the entire circle multiplied by the circumference of the circle. Thus, s = θ π πr or s = θr. 450 Saxon Algebra

16 Example 4 Finding Arc Lengths a. Find the length of arc s 1. Approximate to the nearest tenth. SOLUTION s = rθ s 1 = 5 π = 10π 10.5 cm Write the formula. Substitute s 1 for s, 5 for r, and π Simplify. Use a calculator to approximate. for θ. π s 1 5 cm b. Find the length of arc s. Approximate to the nearest tenth. SOLUTION Step 1: Convert 5 to radians. 5 ( π radians 180 ) = 5π 4 radians Step : Use the formula for arc length. s 5.5 ft s = rθ Write the formula. s =.5 ( 5π 4 ) Substitue s for s,.5 for r, and 5π 4 for θ. =.15π Simplify. 9.8 ft Use a calculator to approximate. Example 5 Application: Planet Rotation Earth makes one complete rotation in 4 hours, and its radius is approximately 955 miles. If an object is fixed on Earth s equator, how far does it travel in 1 hour due to Earth s rotation? 1 4 of a rotation in 1 hour SOLUTION r = 955 mi The central angle that corresponds to a complete rotation is π radians. In 1 hour, Earth makes 1 of a complete rotation. So, θ = 1 π = π s = rθ = 955 π The object travels about 105 miles in 1 hour. Check Find the approximate circumference of Earth: πr π 955 4,850 miles. 1 of 4,850 is slightly greater than 1000, so 105 miles is reasonable. 4 Lesson 6 451

17 Lesson Practice a. Convert 150 to radians. b. Convert - 4π radians to degrees. (Ex 1) (Ex 1) c. Use the unit circle to find the exact value of cos 15. d. Use the unit circle to find the exact value of tan π (Ex ) (Ex ) e. Use a reference angle to find the sine, cosine, and tangent of 10. Find exact values. (Ex ) (Ex 4) 4. f. Find the length of arc s 1. Approximate to the nearest tenth. s 1 4π 8.5 m g. Find the length of arc s. Approximate to the nearest tenth. (Ex 4) 15 8 in. s h. The minute hand of a certain clock is 15 centimeters long. How far does the tip of the minute hand travel in 5 minutes, to the nearest centimeter? (Ex 5) Practice Distributed and Integrated Simplify x + y 4 + xy 1. 4 x - 7 (48) (48) (48). x + (48) x + 1 x x - x Determine if the following probability experiments are binomial experiments. If not, explain why. 4. Rolling a number cube 18 times and recording the results (Inv 5) (Inv 5) 5. Flipping coins 0 times to test if they match half of the time Simplify (4 - ) (40) (40) Expand. 8. (x + 5) 9. (x + 4) (19) (19) 45 Saxon Algebra

18 Solve x - 6 = x = -11x - x (5) (5) 1. Error Analysis Julian is finding the inverse of the function y = x. Which step is (50) incorrect? Step One: y = x Step Two: x = y Step Three: x = y Step Four: ± x = y *1. Write the expression in terms of i and then add: ( ) + ( ). (6) 14. Geometry In the triangle to the right, find x. (44) x m C 90 x m B 45 9 m 45 A *15. Planet Rotation Jupiter is the largest planet. Its radius is approximately 44,65 (6) miles. Jupiter rotates faster than Earth. It takes approximately 9.8 hours for Jupiter to make one complete rotation. If an object is fixed on Jupiter s equator, how far does it travel in 1 hour due to Jupiter s rotation? 16. Multiple Choice Which of the following functions could be used to model (57) exponential growth? A y = 1 B y = e -x e x C y = e x D y =.4 e -x 17. Surveying A surveyor stands on the roof of a building that is 180 feet above the (46) ground and spots the top of a taller building at an angle of elevation of 4. The two buildings are 480 feet apart. Find the height of the taller building. Round to the nearest foot. *18. Graphing Calculator Describe the feasible region for the set of inequalities x 0, (54) y 0, y -1.5x *19. Formulate A fair coin is flipped. Write the equation that can be used to solve for (60) the probability that three flips result in heads, then four flips, and seven flips. Write a formula using exponents that can be used to find the probability that the same independent event will occur n times. 1 x + 1 y 0. Multiple Choice Which of the following expressions is equivalent to (48) 1-1 A 1 - x B x - xy - x - y C x + y (1 - x) x + y x y xy - y D x + y *1. Write How is adding and subtracting complex numbers similar to adding and (6) subtracting polynomials? *. Model Sketch a graph of five points whose correlation coefficient (45) is about 0.1. x? Lesson 6 45

19 180 (n - ) n. Multi-Step The formula θ = gives the measure of each interior angle of (56) an n-sided regular polygon. a. Use the formula to find the measure of an interior angle of a regular decagon. b. Find the reference angle for this angle. 4. Estimate Use what you know about the unit circle and special angles to estimate (6) cos(0.π). Do not use a calculator. Explain your method. 5. Jupiter Gravity on the planet Jupiter is almost three times stronger than on Earth. (58) If an object is thrown into the air, with an initial velocity of 100 feet per second, from the surface of Jupiter, the height, h, of the object in feet after t seconds can be approximated by the equation h = -4t + 100t. Find the time at which the object will first reach 50 feet. Round your answer to the tenths place. 6. Analyze Let f(x) = ax + b and g(x) = cx + d. Under which conditions will the (5) composite functions f(g(x)) and g(f(x)) be equal? 7. Construction The volume of a typical American-made brick in cubic millimeters (61) can be represented by the polynomial x - 0x x + 16,000 where the length is given by x Fully factor the expression for the volume of a brick. 8. Coordinate Geometry Sketch a graph of the equation y = x, where y is the side (59) length of a cube and x is the volume of the cube. Use the graph to estimate the volume of a cube with side lengths of.4 in. Use the graph to estimate the side length of a cube with a volume of 6.9 in. *9. Error Analysis A student incorrectly found s, the length of the (6) indicated arc, as shown below. s = rθ = = 600 m What is the error? Find the correct arc length m s *0. Pass codes A computer generates temporary pass codes for users. Each pass code (55) is exactly letters long with no repeating letters. What is the probability of getting a pass code with vowels (A, E, I, O, or U)? 454 Saxon Algebra

20 LAB 10 Using the Log Keys Graphing Calculator Lab (Use with Lesson # 64, 7, 81, 87, 9, 10, and 110) Evaluating Logarithmic Expressions 1. Calculate the common logarithmic expression 4 log (5). a. To enter the value 4 log (5), press.. b. To calculate this value, press. Calculate the natural logarithmic expression 4 ln (5). a. To enter the value 4 ln (5), press. b. To calculate this value, press Hint Use the Change of Base Law when calculating a log that does not have a base of 10 or e... Use the Change of Base Law to calculate the logarithmic expression log6 (15). a. To enter the value log6 (15), press. b. Press to calculate this value. Graphing Logarithmic Functions Graphing Calculator Tip When graphing logarithmic expressions, use the standard zoom window. Adjust the window to view specific parts of the graph by zooming in or out. 1. Graph the common logarithmic function log (x + ). a. Press to access the screen to enter the logarithmic function. b. To enter the function log (x + ), press., and then press c. Press the ZStandard window. d. Then press to select to view the graph. Online Connection Lab

21 . Graph the logarithmic function log6 (1x) whose base is not natural or common. a. Press to access the screen to enter the logarithmic function. b. To enter the function log6 (1x), press., and then press c. Press the ZStandard window. d. Then press to view the graph. Lab Practice 1. Calculate 6 log (8).. Calculate ln (1).. Calculate log7 (9). 4. Graph y = log (15x + 8). 5. Graph y = log5 (x + 6). 456 Saxon Algebra to select

22 64 LESSON Using Logarithms Warm Up New Concepts 1. Vocabulary In the expression 4 5, the base is 4 and the is 5. (). What is the value of? (). What value of x makes the equation 10 x = 100 true? (47) A logarithm is the exponent that is applied to a specified base to obtain a given value. Every exponential equation has a logarithmic form and vice versa. Exponential Equation Logarithmic Equation b x = a log b a = x b > 0, b 1 The logarithmic equation log b a = x is read the log base b of a equals x. Notice that x is both the exponent and the logarithm in the equations above. Example 1 Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form. a. 5 = Reading Math Read the symbol as if and only if. When used between two equations, it means the equations are equivalent. SOLUTION 5 = log = 5 The base is the same in both the exponential equation and the logarithmic equation. The logarithmic form is log = 5. b. 5 1 = 5 SOLUTION 5 1 = 5 log 5 5 = 1 The exponent is the logarithm. The logarithmic form is log 5 5 = 1. c. 8 0 = 1 SOLUTION Online Connection = 1 log 8 1 = 0 Any nonzero base to the zero power is 1. The logarithmic form is log 8 1 = 0. Lesson

23 Reading Math The equations in Example 1d can be written 5 - = 1 5 and log = -. d. 5 - = 0.04 SOLUTION 5 - = 0.04 log = - An exponent (or log) can be negative. The logarithmic form is log = -. Example Converting from Logarithmic to Exponential Form Write each logarithmic equation in exponential form. a. log = SOLUTION log = 10 = 1000 The exponential form is 10 = b. log 7 7 = 1 SOLUTION The base is the same in both the logarithmic equation and the exponential equation. log 7 7 = = 7 The exponential form is 7 1 = 7. The logarithm is the exponent. c. log 81 = x SOLUTION log 81 = x x = 81 The exponential form is x = 81. The logarithm (and the exponent) can be a variable. A logarithm with base 10 is called a common logarithm. If no base is written for a logarithm, the base is assumed to be 10. For example, log 5 = log Example Application: Acidity of Rainwater Because of the phenomenon of acid rain, the acidity of rainwater is important to environmental scientists. The acidity of a liquid is measured in ph, given by the function ph = -log[h + ], where [H + ] represents the concentration of hydrogen ions in moles per liter. In 1999, the hydrogen ion concentration of rainwater in western Nevada was found to be approximately moles per liter. What was the ph of the rainwater? SOLUTION ph = -log[h + ] ph = -log( ) Substitute the known value in the function. Use a calculator to find the value of the logarithm in base 10. Press the key. The rainwater had a ph of about Saxon Algebra

24 Hint The display e - 6 means , which is approximately. 10-6, or Check -log ( ) 5.5 Write the result found in the Solution. log ( ) -5.5 Multiply both sides by Write the logarithmic equation in exponential form. Use a calculator to verify that the value of is approximately Lesson Practice Write each exponential equation in logarithmic form. (Ex 1) a. = 9 b. 4 1 = 4 c. 9 0 = 1 d. 8-1 = 0.15 Write each logarithmic equation in exponential form. (Ex ) e. log = f. log 8 8 = 1 g. log 5 15 = x h. The acidity of a liquid is measured in ph, given by the function ph = -log[h + ], where [H + ] represents the concentration of hydrogen ions in moles per liter. In 1999, the hydrogen ion concentration of rainwater in northern Maine was found to be approximately moles per liter. What was the ph of the rainwater, to the nearest tenth? (Ex ) Practice Distributed and Integrated Use synthetic substitution. 1. Find f(6) for f(x) = x 4-4x + x + 4x (51). Find f(-7) for f(x) = x 4 + 5x - 1x - 4x + 8. (51). Find the equation of the line that passes through (-, 5) and (-6, -). (6) 4. The sides of a triangle measure 7 in., 6 in., and 8 in. Is the triangle a right triangle? (41) Find the zeros of each quadratic function. 5. f(x) = x + x f(x) = -x - 8x 7. f(x) = x - 9 (5) (5) (5) Determine if (0, -) is a solution of the inequalities. 8. 5x + y < y - x > y - x (9) (9) (9) 11. Formulate A parasail is attached to a boat with a rope 00 feet long. The angle of (5) elevation from the boat to the parasail is 48 degrees. Write a formula that would help you to estimate the parasail s height above the boat. Lesson

25 *1. Coordinate Geometry Find the area of triangle OQR. Give the answer in (6) exact form and to the nearest tenth of a square unit. O(0, 0) y P(, 1 ) Q O x R(, 0) 1. Find an equation for the inverse of y = 5x - 8. Identify the domain and range of the (50) inverse function. *14. Write Explain why log 0 9 and log 1 9 do not exist. (64) 15. Population Currently the U.S. population is growing at a rate of approximately (5) 0.6 percent per year. According to the 000 census, the population was about 81 million. Use composite functions to predict the U.S. population in the year 015, and justify your prediction ) ( ). (59) a. Which property of rational exponents should be applied first to simplify the expression? Explain. b. Apply the property from part a. *16. Multi-Step Use ( 9 c. Which property of rational exponents should be applied next? d. Apply the property from part c. e. If possible, finish simplifying the expression from part d. *17. Transportation The approximate volume of a moving truck, in cubic feet, that can (61) move four to five household rooms can be given by x - 15x + 56x + 6x - 90x + 6. Factor this expression by grouping. Then find the dimensions when x = Geometry The figures below show how to create Sierpinski s Triangle. Start with a (57) solid black equilateral triangle (Iteration 0); at every iteration, join the midpoints of the sides of each black triangle to get a smaller triangle, and color the interior of that triangle white. Iteration 0 Iteration 1 Iteration Iteration Iteration 4 Formulate an exponential function for the number of black triangles N(x) in the x th iteration. 19. Find the slope of the line that passes through the points A ( x, 1 ) and (1) B ( 1 5, 5 x ), where x is any non-zero real number. *0. Windshield Wipers A rear windshield wiper moves through (6) an angle of 15 on each swipe. To the nearest inch, how much greater is the length of the arc traced by the top end of the wiper blade than the length of the arc traced by the bottom end of the wiper blade? Top end Bottom end 14 in. 9 in. 460 Saxon Algebra

26 1. Multiple Choice Which of the following expressions is equivalent to (44)? A B C D Multi-Step What number needs to be added to both sides of the equation to make x + 1x + 4 = 0 a perfect square? Solve the equation. (Inv 6) *. Rainwater The acidity of a liquid is measured in ph, given by the function (64) ph = -log[h + ], where [H + ] represents the concentration of hydrogen ions in moles per liter. In 1999, the hydrogen ion concentration of rainwater in the Chesapeake Bay region of Maryland was found to be approximately moles per liter. What was the ph of the rainwater, to the nearest tenth? Show how to check your answer by writing a logarithmic equation and its equivalent exponential equation. (Hint: You will need to use the symbol instead of the = symbol.) *4. Write Two number cubes are tossed once. Explain why the events, one cube (60) shows a number less than 5 and the sum of the two cubes is a multiple of, are dependent. 5. Justify Give an example of an event that has a probability of 0. Explain why the (55) probability is 0. *6. Graphing Calculator Set the mode on your graphing calculator to a + bi by pressing (6) the Mode key and using the arrow keys. Then find Error Analysis Rizwan tried to find the value of cosθ, where θ is an angle in standard (56) position with the point Q(-6, 8) on its terminal side. His work is shown below. What was Rizwan s error? r = x + y r = (-6) + 8 r = 100 r = 10 cosθ = x r cosθ = 6 10 = 0.6 *8. Multiple Choice Which equation is equivalent to log 16 = x? (64) A x = 16 B x = 16 C 16 = x D 16 = x 9. Analyze Compare and contrast correlation coefficient values of and (45) *0. Falling Objects The height, in feet, of an object that is falling or is projected into (58) the air can be described by h = -16t + v 0 t + h 0, where h is the height in feet after t seconds, v 0 is the initial velocity of the object in feet per second, and h 0 is the initial height of the object in feet per second. A coin is tossed from the top of a 100 foot tall building, with an initial velocity of 9 feet per second. Write the equation that models the height of the coin. Find the time when the coin will reach the ground. Round your answer to the tenths place. Lesson

27 65 LESSON Using the Quadratic Formula Warm Up New Concepts Math Reasoning Analyze Explain how -c a + b 4a 4ac becomes - 4a + b 4a. 1. Vocabulary The square root of a negative number is an number.. Solve by completing the square x - 6x = 1. (6) (58). Simplify -5. (6) 4. Evaluate b - 4ac when a =, b = 7, and c = 5. () In general, a quadratic equation in standard form is ax + bx + c = 0, with a 0. The general equation can be solved for x by completing the square. x + b ax + bx = -c x + b a x = - c a a x + b 4a = - c ( x + b a ) = - c ( x + b 4ac ( x + b a + b 4a a + b 4a a ) = - 4a + b 4a a ) = b - 4ac 4a x + b a = ± b - 4ac 4a b x = - a ± b - 4ac Isolate x. 4a b x = - a ± b - 4ac a x = -b ± b - 4ac a The last line is the quadratic formula. Subtract c from each side. Divide both sides by a. Complete the square. Factor the left side. Make common denominators on the right. Combine the fractions on the right. Apply the Square Root Property. Simplify the radical. Combine the fractions. Quadratic Formula The solutions of the quadratic equation ax + bx + c = 0 (a 0) are given by x = -b ± b - 4ac. a Online Connection The quadratic formula can be used to find the solutions of any quadratic equation. 46 Saxon Algebra

28 Example 1 Solve each equation. Solving Quadratic Equations with Real Zeros Math Language The solutions of a quadratic equation are the zeros of the related function. a. 5x + 4x - 7 = 0 SOLUTION Use the quadratic formula. x = -b ± b - 4ac a = -4 ± 4-4(5)(-7) (5) 196 = -4 ± 10-4 ± 6 = 10 The solutions are 1 5 and -7. Substitute 5 for a, 4 for b, and -7 for c. Simplify the radicand and denominator. = 1 and -7 Evaluate the square root and simplify. 5 b. 9x + 6x + 1 = 0 SOLUTION Use the quadratic formula x = -b ± b - 4ac. a x = -6 ± 6-4(9)(1) (9) 0 = -6 ± 18 = -6 ± 0 18 The solution is - 1. = = - 1 Substitute 9 for a, 6 for b, and 1 for c. Simplify. Example Solving Quadratic Equations with Complex Zeros Hint The standard form of a complex number is a + bi. a. Solve the equation x + 5x + 4 = 0. Write the solutions as complex numbers in standard form. SOLUTION Use the quadratic formula. x = -b ± b - 4ac a = -5 ± 5-4()(4) () - = -5 ± 6 = -5 ± i 6 Substitute for a, 5 for b, and 4 for c. Simplify the radicand and denominator. Write the negative root as an imaginary number. = ± i Write the solutions in standard form. 6 The solutions are i and i. 6 - Lesson 65 46

29 b. Solve the equation x - 1 x + 1 = 0. Write the solutions as complex 8 numbers in standard form. SOLUTION To make the calculations simpler, clear the fractions before using the quadratic formula. 8x - 4x + 1 = 0 Multiply both sides of the equation by 8. Use the quadratic formula: x = -b ± b - 4ac. = -(-4) ± (-4) - 4(8)(1) (8) = 4 ± = 4 ± 4i 16 = 1 ± i 4 a Substitute 8 for a, -4 for b, and 1 for c. Simplify the radicand and denominator. Write the negative root as an imaginary number. Factor 4 out of the numerator and denominator. = 1 4 ± 1 i Write the solutions in standard form. 4 The solutions are i and i. Because the solutions of a quadratic equation are the zeros, or x-intercepts of the equation, a graphing calculator can be used to find the solutions of a quadratic equation when the solutions are real. Math Reasoning Verify Graph y = x + 5x + 4. How does the graph confirm that there are no real solutions to x + 5x + 4 = 0? Example Solving Quadratic Equations using a Graphing Calculator Solve x + x - 15 = 0 by using a graphing calculator. SOLUTION Graph the related function, y = x + x If needed, adjust the window so that the x-intercepts of the graph are displayed. Access the Calc function (nd + Trace) and choose zero. When asked for left and right bounds, choose points that are on either side of one of the x-intercepts. Repeat for the second intercept. The solutions are - and Saxon Algebra

30 Example 4 Application: Travel Train A leaves the station at 1 p.m. and travels due north at 55 miles per hour. Train B leaves the same station at 4 p.m. and travels due east at 5 miles per hour. About how long will it take for the trains to be 500 miles apart? SOLUTION Write each distance as rate time. Train B is hours behind Train A. Train A: d = rt = 55t Train B: d = rt = 5(t - ) 55t 500 a + b = c (55t) + (5(t - )) = t + (5t - 105) = 50,000 05t + 15t - 750t + 11,05 = 50, t - 750t - 8,975 = 0 Now use the quadratic formula. t = -(-750) ± (-750) - 4(450)(-8,975) (450) = 750 ± 4,116,597, t 8.41 and t Use the Pythagorean Theorem. Simplify. Square the binomial. Combine like terms. Disregard the negative time. It will take about 8.41 hours for the trains to be 500 miles apart. Check Find the distance each train traveled after 8.41 hours. Train A: 55(8.41) = miles, Train B: 5( ) = miles Use the Pythagorean Theorem: = (t - ) Lesson Practice Solve each equation. (Ex 1) a. x - 7x + = 0 b. 9x + 1x + 4 = 0 Solve each equation. Write the solutions as complex numbers in standard form. c. x + 5x = - 9 (Ex ) d. -x + 1 x = 0 (Ex ) (Ex ) (Ex 4) e. Solve 8x + 7x = 0 by using a graphing calculator. f. A hiker leaves a campground at 8 a.m. and heads due south at a rate of 4 miles per hour. Another hiker leaves the same campground at 9 a.m. and heads due east at a rate of miles per hour. Approximately how long will it take for the hikers to be 15 miles apart? Lesson

31 Practice Distributed and Integrated 1. Secret ingredient X has a half-life of hours. What portion of 60 kg will remain after 7 hours? (57) Use long division.. Divide (x + 1x - 15x + 15) by (x - 9) (8). Divide (x + 14x - 4x - 48) by (x + 4) (8) Solve. *4. x = 7 + x *5. x = -x + 1 (58) (58) Simplify. 6. (5-6 ) (40) (40) (40) 9. Jordon spins a spinner with six equal-sized sections numbered 1 6. In one spin, what is the likelihood that the spinner will stop on a 1 or a 5? (55) *10. Error Analysis Find and correct the error a student made in solving 4x - 7x + 9 = 0. (65) x = -b ± b - 4ac a = -7 ± (-7) - 4(4)(9) 8 = -7 ± ± 15 = = 8-7 ± 5 = - 1 and List the coefficients in the 6 th row of Pascal s Triangle. (49) 1. Probability A history test has 0 questions, 7 of which are about the Civil War. (55) Students are asked to answer 5 of the 0 questions. What is the probability that a student randomly selecting questions will select 5 Civil War questions? *1. Write Describe how to determine sin π (6) by using the unit circle. 14. Find an equation of a line perpendicular to y = 1 x - 4. (6) 15. Analyze In what type of right triangles will the sine and cosine values of the acute (46) angles be equal? Find x and y (5) (5) y x 5 y 0 x Saxon Algebra

32 *18. Multi-Step Complete parts a d to graph a logarithmic function and use the graph (64) to estimate a logarithm. a. Write the equation y = log 1 4 in exponential form. What value of y makes both equations true? b. Writing exponential equations if needed, complete the table of values for the function y = log x. x y = log x c. Plot the points and sketch the graph of the function. d. Use your graph to estimate the value of log 6. Check your estimate by evaluating the appropriate power of on a calculator. 19. Multiple Choice Choose the letter that best represents the value of x in the equation (58) x + x + 7 = 8. A x = -1 ± B x = -, 0 C x =, 0 D x = 1 ± 4 0. Measurement Given that 1 strip of red ribbon = inches and 1 strip of blue ribbon (9) = 4 inches, write and graph the inequality that represents that the total number of inches must be no less than Multi-Step A car is sitting outside in a driveway. Consider the following statement, If it starts raining, the car will get wet. a. Is the converse of the statement true? Explain. b. Is the contrapositive true? Explain. (Inv 1). Geometry The base of a triangle is 5 feet, and its height is 1 (44) area of the triangle. feet. Find the *. Physics A fireworks technician wants a firework to explode 70 feet below the (6) top of the Gateway Arch in St. Louis. The Gateway Arch has a height of 60 feet. The technician has figured the height of the firework to be modeled by -16t + 160t, where t is the time in seconds after it is launched. The solutions of -16t + 160t = 560 give the number of seconds it will take for the firework to reach the desired height. Solve the equation. Will the firework reach the desired height? If so, when? 4. Justify Let f(x) = x - 7 and g(x) = 1 x + 7 (5) a. Justify the claim that f (g(x)) = x. b. Does g(f(x)) = x also? Explain. 5. Error Analysis Find and correct the error a student made in factoring the expression (61) below.. 9x 4-6x - 8x x (x - 7) - 8(x - 7) (x - 7)(9x - 8) (x - 7)(x - )(6x + 6x + 4) Lesson

33 *6. Physical Science The North Falls waterfalls in Silver Falls State Park in Oregon are (65) 16 feet high. The equation -16t + 16 = 0 gives the number of seconds it takes for a drop of water at the top of the falls to reach the pool of water below. Solve the equation by using the quadratic formula. Round to the nearest hundredth of a second. 7. Formulate Let θ be an angle in standard position and let n be any integer. You can (56) use the expression θ + n(60 ) to indicate all the angles coterminal with θ. Write an expression for all the angles coterminal with 15, and find different ones. *8. Graphing Calculator Solve 5x - x - 16 = 0 by using a graphing calculator. (65) 9. Velocity of an Electron The velocity at which an electron orbits the nucleus of (59) an atom is known as the Fermi-Thomas velocity and can be found using the equation Z 17 c, where Z is the number of protons in the nucleus and c is the speed of light. What is the Fermi-Thomas velocity for platinum, which has 78 protons, in terms of c? Round to the nearest thousandths place. *0. Analyze Write the following equations in exponential form. What property of (64) exponents is illustrated? log 1 = 0, log 1 = 0, log 15 1 = Saxon Algebra

34 66 LESSON Solving Polynomial Equations Warm Up New Concepts Hint Recall that the Zero Product Property states that if ab = 0, then a = 0 or b = Vocabulary The formula x = -b ± b - 4ac is called the. a. Factor 7cd - 1cd. (65) (). Solve x(x - 4) = 0. (5) 4. True or False: x + 7 is a factor of x + 8x + x - 4. (61) 5. Solve x + x + 1 = 0 by using the quadratic formula. (65) Sometimes a polynomial equation can be solved just by factoring and using the Zero Product Property. Example 1 Solve each equation. a. x 4-10x - 7x = 0 Using Factoring to Solve Polynomial Equations SOLUTION Since one side is already equal to 0, begin by factoring the polynomial. x (x - 5x - 6) = 0 The GCF is x. x (x + 4)(x - 9) = 0 x = 0 or x + 4 = 0 or x - 9 = 0 Factor the trinomial. Use the Zero Product Property. x = 0 or x = -4 or x = 9 Solve each equation. The solutions are 0, -4, and 9. b. 4x + x = 4x SOLUTION First write the equation in standard form. 4x - 4x + x = 0 x(4x - 4x + 1) = 0 The GCF is x. x(x - 1)(x - 1) = 0 x = 0 or x - 1 = 0 or x - 1 = 0 x = 0 or x = 1 The solutions are 0 and 1. or x = 1 Subtract 4x from each side. Factor the perfect square trinomial. Use the Zero Product Property. Solve each equation. Online Connection Notice in Example 1b that the solution, or root, of 1 called a double root. It has multiplicity. appears twice. This is Lesson

35 Multiplicity of Roots The multiplicity of a root r of a polynomial equation is the number of times x - r is a factor of the polynomial. Example Identifying Multiplicity Find the roots of each equation. Give the multiplicity of each root. a. x 5 + 8x x = 0 SOLUTION Factor the polynomial to use the Zero Product Property. x (x + 8x + 16) = 0 The GCF is x. x (x + 4)(x + 4) = 0 x = 0 or x + 4 = 0 or x + 4 = 0 Factor the perfect square trinomial. Use the Zero Product Property. x = 0 or x = -4 or x = -4 Solve each equation. The roots are 0 and -4. Because x = x x x, x is a factor times, so the root of 0 has a multiplicity. Because x + 4 is a factor times, the root of -4 has multiplicity. b. x 4 = x - 1 SOLUTION First write the equation in standard form: x 4 - x + 1 = 0. (x - 1)(x - 1) = 0 Factor the perfect square trinomial. (x + 1)(x - 1)(x + 1)(x - 1) = 0 Factor the differences of squares. x + 1 = 0 or x - 1 = 0 or x + 1 = 0 or x - 1 = 0 x = -1 or x = 1 or x = -1 or x = 1 The roots are -1 and 1. Because both (x + 1) and (x - 1) occur twice, each root has multiplicity. Math Reasoning Analyze What can you do to use the Rational Root Theorem to find the possible rational roots of 1 5 x + x - x + = 0? A polynomial equation like x - x - 11x + 1 = 0, where the degree of the polynomial is greater than, can be factored by using synthetic division if one of the factors is a linear binomial and is already known. When none of the factors are already given, begin factoring the polynomial by using the Rational Root Theorem. The Rational Root Theorem If a polynomial P(x) has integer coefficients, then every rational root of P(x) = 0 can be written in the form p q, where p is a factor of the constant term and q is a factor of the leading coefficient of P(x). The Irrational Root Theorem states that irrational roots come in pairs. The Irrational Root Theorem If a polynomial P(x) has rational coefficients, and a + b c is a root of P(x) = 0, where a and b are rational and c is irrational, then a - b c is also a root of P(x) = Saxon Algebra

36 Example Identifying All Real Roots of a Polynomial Equation a. Identify all the real roots of x - x - 11x + 1 = 0. Hint Remember that x - r is a factor if P(r)= 0. SOLUTION By the Rational Root Theorem the possible rational roots ±1, ±, ±, ±4, ±6, ±1 are: = ±1, ±, ±, ±4, ±6, ±1. Use synthetic ±1 division to test roots, r, until a factor, x - r, is found The first try works. By the Factor Theorem P(1) = 0, so x - 1 is a factor. The quotient is also a factor, so the polynomial can be written as (x - 1)(x - x - 1), which factors into (x - 1)(x - 4)(x + ). The equation can be written as (x - 1)(x + )(x - 4) = 0. Using the Zero Product Property, the roots are 1, -, and 4. b. Identify all the real roots of x 4-5x - 9x + x + 4 = 0. SOLUTION By the Rational Root Theorem the possible rational roots are: ±1, ±, ±4 ±1, ± = ±1, ± 1, ±, ±, ±4, ± 4. Test roots. Begin with integers. Organize the work in a synthetic substitution table. The remainder is 0 for P(4), so x - 4 is a factor. x + 7x - x - 1 is the other factor. Possible root The possible rational roots for x + 7x ±1 - x - 1 = 0 are: ±1, ± = ±1, ± x is a factor. x + 6x - is the other factor. The trinomial factors into (x + x - 1). Solve -1-1 x + x - 1 = 0 by using the quadratic formula. 6-0 x = - ± -4(1)(-1) (1) The polynomial equation is = - ± 8 = - ± (x - 4) ( x + 1 ) (x - (-1 + ) ) (x - (-1 - ) ) = 0. = -1 ± The real roots are 4, - 1, -1 +, and The x-intercepts of a polynomial function are real roots of the related equation. Therefore, a graph can be used to help find the roots of a polynomial equation. Lesson

37 Example 4 Using a Calculator to Solve Polynomial Equations Solve x + 5x - 9x - 10 = 0 by using a graphing calculator. SOLUTION Graph the related function, y = x + 5x - 9x Use the Calc command to find the x-intercepts. Math Reasoning Verify Show that the Rational Root Theorem gives as a possible root. Since is a rational root, use synthetic division with r = Solve x + 7x + 5 = 0: x = -7 ± 7-4(1)(5) (1). = -7 ± 9 The roots are, , and Notice that the x-intercepts given by the graphing calculator above are estimates of the exact values of the irrational roots. Example 5 Application: Packaging A box is to be designed so that the height is 6 inches greater than the length and the width is inches less than the length. What must the length be if the volume has to be 40 cubic inches? SOLUTION Hint Use the Rational Root Theorem to determine all the possible rational roots. 1. Understand The volume of a box, or rectangular prism, is the product of the length, width, and height: V = lwh.. Plan Write expressions for each dimension. If x represents the length, then x + 6 represents the height and x - represents the width. 40 = x(x - )(x + 6). Solve 40 = x + x - 18x 0 = x + x - 18x The polynomial factors into(x - 4)(x + 7x + 10), which factors into (x - 4)(x + )(x + 5). The roots are 4, -, and -5. Since the length must be positive, disregard the negative roots. The length is 4 inches. 4. Check If the length is 4 inches, the height is 10 inches, and the width is 1 inch. Find the volume: V = lwh = 4(1)(10) = Saxon Algebra

38 Lesson Practice Solve each equation. (Ex 1) a. x 4 - x - 8x = 0 b. 18x - 18x = 0 Find the roots of each equation. Give the multiplicity of each root. (Ex ) c. x - x + 18x = 0 d. x 6 + 1x 5 + 7x 4 = 0 (Ex ) e. Identify all the real roots of x - 5x - x + 5 = 0. f. Identify all the real roots of x + 4x - 5x - 14 = 0. g. Solve x + x - 45x - 18 = 0 by using a graphing calculator. h. A box is to be designed so that the height is inches greater than the length and the width is inches less than the length. What must the length be if the volume has to be 18 cubic inches? (Ex ) (Ex 4) (Ex 5) Practice Distributed and Integrated Simplify each expression p - 4px (59) p 4 (7) x x x x + 4x a a a + a 5. (4 6 ) *7. There are 7 cherries and grapes in a bag. What is the probability of selecting (60) grapes at random without replacement? 8. Find the equation of the line that passes through (, -) and is parallel to y = - 8 x +. *9. Packaging A new food product is to be sold in cylindrical cans. The designer wants (66) the height of the can to be 6 inches greater than the radius. The manager wants the can to have a volume of about 15 in. a. Write a polynomial equation in standard form that can be used to find the radius. Leave in terms of π. Hint: The formula for the volume of a cylinder is V = πr h. b. Find the approximate radius that should be used for the can by using a graphing calculator. Round the radius to the nearest tenth. 10. Find an equation for the inverse of y = 7x + 1 for x 0. Identify the domain (50) and range of the inverse function. 11. Multiple Choice f(x) = x - 5 is an example of a: () A continuous function B discrete function C discontinuous function (7) (7) (7) (59) (59) (6) D both B and C Lesson 66 47

39 1. Write Explain why it is necessary to add the constant term ( b ) to both sides of (58) the equation when using the method of completing the square to solve a quadratic equation. *1. Graphing Calculator Calculate the binomial probability, to the nearest hundredth of (Inv 5) a percent, of 8 successes out of 50 trials, if the probability of success for each trial is Geometry Consider the circle with radius 5 units, centered at the origin. A 70 (56) angle in standard position cuts the circle into a larger and a smaller sector. Find the area of the larger sector. *15. Error Analysis A student converted the logarithmic equation log 64 = x to (64) exponential form by writing x = 64. What is the error? Write the correct exponential equation. *16. Multi-Step A 4 in. by 6 in. photo is framed and the total area of the photo and (65) the frame is 100 in. a. Write a quadratic expression in standard form to represent the situation. b. Find the width of the frame to the nearest hundredth of an inch. 17. Formulate Write the equation of an exponential function whose graph passes (57) through the points (0, 5) and (-, 15). Determine whether the function is increasing or decreasing. *18. Analyze If using the quadratic formula to solve 16 = -x, what would be the (65) values of a, b, and c? *19. Tires The diameter of a certain tire is 4 inches. Through how many degrees does (6) the tire rotate when the automobile travels 5 feet? Give your answer to the nearest degree. 0. Population In 006, the U.S. Census Bureau estimated there were 6,457,549 (55) people living in California and 99,98,484 living in the United States. a. Estimate the probability that a person randomly selected in the United States lives in California. b. Estimate the odds in favor that a person randomly selected in the United States lives in California. 1. Probability A student taking a test has enough time to solve out of the (6) 5 equations shown. x + 1 = 0, x + 4 = 0, x - 5 = 0, x - 16 = 0, x + 9 = 0 If she chooses equations randomly, what is the probability that both equations have imaginary solutions? Use trigonometric ratios to solve for each of the variables.. cos 0 = and cos 0 = y. sin 0 = 1 1 and sin 0 = x (5) (5) Saxon Algebra

40 4. Multi-Step A student creates a chart classifying (60) his books by subject and cover. a. Are the events math textbook or paperback textbook mutually exclusive or inclusive? Explain why. b. Find the probability that four books, selected randomly with replacement, are math or paperback. Round to the thousandths place. Cover Subject Hardcover Paperback Math 7 Sciences 14 Social Sciences 6 16 History Music Frets are small metal bars placed across the neck of a guitar so that the (59) guitar can produce notes of a specific scale. To find the distance a fret should be placed from the bridge, multiply the length of the string by - n 1, where n is the number of notes higher than the string s root note. Where should a fret be placed on the low E string to produce a note that is 5 notes higher, if the length of the string is 64 cm, rounded to the nearest hundredths place? *6. Generalize Tell how to factor by grouping. (61) 7. Probability Find the probability of getting at least 1 tail if a coin is tossed 4 times. (49) *8. Multiple Choice Which is not a possible rational root of x + x - 1x - 7 = 0? (66) A - B -1 C 9. Let f(x) = 8x - 6 and g(x) = 4x + 1. Evaluate the composition (f g)(-) using (5) two different methods. *0. Write Explain why log 0 5 and log 1 5 do not exist. (64) D 9 Lesson

41 67 LESSON Finding Inverse Trigonometric Functions Warm Up New Concepts Reading Math The symbol sin -1 represents the inverse sine relation, NOT a reciprocal. sin -1 a 1 sin a. 1. Vocabulary A circle with a radius of 1 unit is called a. (6). What is sin 0?. What is cos π (6) (6) 4? Evaluation of trigonometric functions for given angle measures was explained in Lesson 46. Angle measures that have given trigonometric function values can be found by using inverse trigonometric relations. If a function value is: Then one of the inverse relation values is: sin θ = asin -1 a = θ cos θ = acos -1 a = θ tan θ = atan -1 a = θ The inverses of the trigonometric functions are not functions themselves because there are many values of θ for a particular value of a. For example, consider cos -1. Angles with measures of π 4 and 7π 4 radians have the cosine value. So do all angles that are coterminal with these angles. 7π 4 y π 4 Unit Circle (, ) x (, ) Hint Substitute positive and negative integers for n in the expressions π + πn and π + πn. Evaluate the sine of each result. Example 1 Finding Trigonometric Inverses Describe all the values of sin -1. Step 1: Find the values of θ between 0 and π radians for which sin θ is equal to. ( 1, = sin π, = sin π Use y-coordinates of points on the unit circle. Step : Describe the measures of all angles that are coterminal with π and π. ) π y π ( 1, ) x All the values of sin -1 are given by π + (π)n and π + (π)n, where n is an integer. Add integer multiples of π radians. Online Connection To define inverse trigonometric functions, it is necessary to restrict the domains of the trigonometric functions. The trigonometric functions that a calculator uses have restricted domains and will give only one value in the range that you will need to generalize to find all the possible values. 476 Saxon Algebra

42 Sin θ = sin θ for θ - π θ π θ can terminate on the y-axis, the positive x-axis, or in Quadrant I or IV. π x Cos θ = cos θ for {θ 0 θ π} Tan θ = tan θ for θ - π < θ < π θ can terminate on the x-axis, the positive y-axis, or in Quadrant I or II. θ can terminate on the positive x-axis, or in Quadrant I or IV. Sin θ, Cos θ, and Tan θ can be used to define the inverse trigonometric functions. For each value of a in the domain of any inverse trigonometric function, there is one and only one value of θ in the range. π - π - π - π π x x Reading Math The inverse trigonometric functions are also called the arcsine, arccosine, and arctangent functions. Inverse Trigonometric Functions Words Symbol Domain Range The inverse sine function is Sin -1 a = θ, Sin -1 a {a -1 a 1} θ - π θ π {θ -90 θ 90 } where Sin θ = a. The inverse cosine function is Cos -1 a = θ, where Cos θ = a. The inverse tangent function is Tan -1 a = θ, where Tan θ = a. Cos -1 a {a -1 a 1} {θ 0 θ π} {θ 0 θ 180 } Tan -1 a {a - < a < } θ - π < θ < π {θ -90 < θ < 90 } Example Evaluating Inverse Trigonometric Functions Find each inverse trigonometric function value that exists. Give answer in both radians and degrees. y a. Tan -1 (-1) Hint Tan θ = y x Tan π (- 4 ) = - = -1 SOLUTION Use the unit circle. -1 = Tan ( - π 4 ) Tan -1 (-1) = ( - π 4 ), or Ta n -1 (-1) = -45 b. Co s -1 SOLUTION - π is the angle measure between 4 - π and π whose tangent is -1. x,or 45 4 (, ) The domain of the inverse cosine function is {a -1 a 1}. Because is not in this domain, Co s -1 does not exist. π Lesson

43 Example Solving Trigonometric Equations Caution The inverse cosine key on a calculator will only give the solution in the domain of the inverse cosine function. Cos -1 = 0 a. Solve cos θ =, for 0 θ 60. SOLUTION cos θ = cos θ is unknown. cos θ = cos θ = Divide each side by to isolate cos θ. 0 y (, 1 ) 0 x (, 1 ) θ = 0 or 0 Use the unit circle. The solutions are 0 and 0. b. Solve 5 sin θ = 5, for 0 θ 60. SOLUTION (0, 1) y 5 sin θ = 5 sin θ is unknown. sin θ = 1 Divide each side by 5. θ = 90 Use the unit circle. The solution is x c. Solve 6 sin θ + = 0, where θ is any real number of radians. SOLUTION 6 sin θ + = 0 sin θ is unknown. 6 sin θ =- Subtract from each side. sin θ =- 1 Divide each side by 6. θ = 7π or 11π or any coterminal angle ( 6 6 The solutions are 7π 11π + πn and 6 6, 1 7π 6 + πn, where n is an integer. y 11π 6 ) ( x ), 1 D Example 4 Solving Trigonometric Equations Using a Calculator Solve each equation to the nearest tenth. Use the given restrictions. Caution If the answer on your calculator screen is when you enter co s -1 (0.6), your calculator is set to radian mode instead of degree mode. a. 10 Cos θ = 6, for 0 θ 180 SOLUTION 10 Cos θ = 6 Cos θ is unknown. Cos θ = 0.6 Divide each side by 10. θ = Co s -1 (0.6) Use the inverse cosine key on your calculator. θ 5.1 The solution is Saxon Algebra

44 b. 10 cos θ = 6, for 70 < θ < 60 SOLUTION Because of the restrictions, the terminal side of θ is in Quadrant IV. Find the angle that terminates in Quadrant IV that has the same cosine value as 5.1. θ θ 06.9 y 06.9 x 5.1 θ has a reference angle of 5.1, and 70 < θ < 60 Example 5 Application: Navigation A pair of hikers plan to walk from a campground to a waterfall. The waterfall is.5 miles east and 0.5 mile north of the campground. To the nearest degree, in what direction should the hikers head? Step 1: Draw a diagram. The hikers direction should be based on θ, the measure of an acute angle of a right triangle. N 0.5 Waterfall Campground θ.5 Step : Find the value of θ. Tan θ = opposite Use the tangent ratio. adjacent Tan θ = = 0. θ = Ta n θ 11 Substitute 0.5 for opposite and.5 for adjacent. Then simplify. The hikers should head 11 north of east. Lesson Practice a. Describe all the values of co s -1 (Ex 1) b. Evaluate Co s -1 (- ) in both radians and degrees if it exists c. Evaluate Si n -1 (- ) in both radians and degrees if it exists. d. Solve sin θ =-1, for 0 θ 60. (Ex ) (Ex ) (Ex ) (Ex ) (Ex ). e. Solve -cos θ =-1, for 0 θ 60. f. Solve 10 cos θ - 5 = 0, where θ is any real number of radians. Lesson

45 g. Solve the equation to the nearest tenth. Use the given restrictions. 5 Sin θ =, for - 90 θ 90 (Ex 4) h. Solve the equation to the nearest tenth. Use the given restrictions. 5 Sin θ =, for 90 < θ < 180 (Ex 4) (Ex 5) i. A group of hikers plan to walk from a parking lot to an overlook. The overlook is 1.6 miles west and 0.6 mile north of the parking lot. To the nearest degree, in what direction should the hikers head? Practice Distributed and Integrated 1. Given θ = -1787, find the measure of the reference angle. (56) Convert each measure from degrees to radians (6) (6) (6) 5. Use synthetic substitution to find f(-) for f(x) = 7x 4 - x + x (51) 6. Complete the square and factor the resulting perfect square trinomial: x + 6x. (58) Simplify. 7. a b a + b (48) (48) b 8. s a + b + x b a s + x *9. Error Analysis A student incorrectly solved the equation sinθ = -0.55, for (67) 90 θ 70, as shown below. What error did the student make? Find the correct solution. sin θ = si n -1 (-0.55) -.4 θ θ *10. Tennis A tennis player 6 feet from the net hits (67) an overhead shot, hitting the ball at a height of 8.5 feet. Assume that the path of the ball is a straight line. To the nearest degree, what is the maximum angle θ at which the ball can be hit and land within the court? θ 9 ft 45 ft *11. Graphing Calculator a. Graph y = x 4 + x - 1x + 14x - 5 on a graphing (66) calculator. What are the x-intercepts? b. Find the multiplicity of each root of x 4 + x - 1x + 14x - 5 = 0 by factoring. 480 Saxon Algebra

46 1. Minimize P = 4x + 6y for the constraints (54) 0 x 4 y 1 y -x + 4 *1. Multiple Choice Which is a solution of 5 + x = -x? (65) A 1 B 5 C D i 4 *14. Contact Lenses The contact lens prescription for Asia s left eye is stronger than the (Inv 5) prescription for her right eye. If Asia accidentally mixed up her contact lenses, what is the probability that she will put the correct lens in either eye on the first try? Explain. *15. Multi-Step Complete parts a d to determine the measure of the angle formed (6) by the hands of a clock when the time is :4. Find all measures in radians. a. Find x, the angle formed at :00. b. Find (x + y + z), the angle through which the minute hand moves in 4 minutes. x z y c. Find y, the angle through which the hour hand moves in 4 minutes. d. Find z, the angle formed by the minute hand and hour hand when the time is : Model Make a Venn Diagram to illustrate how the complex numbers, real numbers, (6) and imaginary numbers are related. 17. Finance The value of a stock over the past five years where x is the number of (61) years is modeled by x - x + 5x Factor this expression. 18. Geometry The Community Center wants to build a playground area and enclose it (54) within a wooden, split rail fence. The cost of fencing is $0 per finished foot. The cost for painting the fence is $1 per finished foot. The Center cannot spend more than $000 on fencing. The playground cannot be any longer than 0 ft and needs to be at least 15 ft wide. What is the maximum area the Center can get for the size of the playground? How much will this cost? 19. Error Analysis A cup contains 15 pink beads, 4 purple beads, and 18 gold beads. Three (60) beads are individually selected with replacement. The work below is used to find the probability that one of each color bead is selected in the order pink, purple, gold. Is the work correct? If not, explain the error and find the correct solution. If so, explain each step of the work. K = event of pink bead drawn P(K,U,G ) = P(K ) P(U ) P(G ) U = event of purple bead drawn G = event of gold bead drawn P(K ) = = 5 19 = P(U ) = 4 57 = 8 19 P(G ) = = Multiple Choice Choose the letter that best represents 16x1 4 simplified. (59) x A 4 4 B x 4 C 6x 4 x 4 D 4 Lesson

47 *1. Analyze When solving x + 7x + 7x - 6 = 0, why is it not necessary to use (66) synthetic substitution after one of the roots is found?. Write A ball is tossed with an initial velocity of 74 feet per second from the top of (58) a building that is 50 feet tall. The height, h, of the ball in feet after t seconds can be approximated by the equation h = -16t + 74t The ball will be 0 feet from the ground when t = second and t = 5 seconds. Explain why you would not use the solution t = second. *. Coordinate Geometry Find m AOB to the nearest tenth of a degree, using these (67) points: A(8, ), O(0, 0), and B(8, 0). 4. Verify Explain why the probability of choosing first, second, and third place (55) winners for a photo when randomly choosing from the top 10 athletes is 1. 10C *5. Storage The volume of a can of dog food is 450 cubic inches. The height of the (66) can is 5 inches greater than its radius. Write a polynomial in standard form, using.14 for π, that can be used to find the radius. (Round the constant to the nearest integer.) Then find the approximate radius by using a graphing calculator. 6. Generalize Suppose θ is an angle in standard position and Q(x, y) is a point on the (56) terminal side of the angle. Explain how the signs (positive or negative) of sin θ, cos θ, and tan θ depend on which quadrant Q lies in. 7. School A teacher has red crayons, 7 yellow crayons, 5 green crayons, and blue (60) crayons. A yellow crayon and then a blue crayon are chosen at random without replacement. Are these events dependent or mutually exclusive? 8. The number of rabbits increases exponentially. At first there were 400. Three (57) years later there were How many rabbits would there be after 10 years? 9. Maximize P = 5x + y for the constraints (54) y 0 x 0 y -x + 10 y x + 1 *0. Physical Science The South Falls waterfalls in Silver Falls State Park in Oregon (65) are 177 feet high. The equation -16t = 0 gives the number of seconds it takes for a drop of water at the top of the falls to reach the pool of water below. Solve the equation by using the quadratic formula. Round to the nearest hundredth of a second. 48 Saxon Algebra

48 68 LESSON Finding Conditional Probability Warm Up 1. Vocabulary If the occurrence of one event affects the probability of the second event, the two events are. (55). As a percent, what is the probability of getting heads on all three flips of a coin? (55). A bag contains 8 red marbles, 16 white marbles, and 4 black marbles. What is the probability of randomly selecting a white or black marble? (55) New Concepts The probability of two independent events is the product of their individual probabilities. However, finding the probability of two dependent events requires finding a conditional probability. Recall that P(B A) is a conditional probability. It is read as the probability of B given that A has occurred. Dependent Events If A and B are dependent events, then P(A and B) = P(A) P(B A). Example 1 Finding the Probability of Dependent Events Math Reasoning Analyze Name another way of finding the probability of the events in Example 1a. a. Suppose the two spinners shown are each spun once. Find the probability that the first spinner lands on and the sum of the spinners is 6. SOLUTION Assign event A to be the first spinner landing on and event B to be the sum of the spinners is 6. Determine if the events are dependent. It helps to list all the outcomes in the sample space. The first numbers are from the first spinner, and the second numbers are from the second spinner. If event A does not occur, P(B) = because out of 16 possible outcomes, have a sum of 6. These 16 pairs are boxed. If event A does occur, P(B) = 1 because out of 4 possible 4 pairs only one has a sum of , 1, 1, 1 4, 1 1,,, 4, 1,,, 4, 1, 4, 4, 4 4, 4 1, 1, 1, 1 4, 1 1,,, 4, 1,,, 4, 1, 4, 4, 4 4, 4 Online Connection Since the occurrence of event A affects the probability of event B, the events are dependent. P(A and B) =P(A) P(B A) = = 1 16 Lesson 68 48

49 b. Suppose the two spinners shown are each spun once. Find the probability that the first spinner lands on an odd number and the sum of the spinners is greater than SOLUTION Confirm that the events are dependent. If event A does not occur, P(B) = = 5 because out of 16 possible 8 outcomes, 10 have a sum greater than 4. If event A does occur, P(B) = 4 8 = 1 because out of 8 possible pairs (those in the first and third columns), 4 have a sum greater than 4. 1, 1, 1, 1 4, 1 1,,, 4, 1,,, 4, 1, 4, 4, 4 4, 4 1, 1, 1, 1 4, 1 1,,, 4, 1,,, 4, 1, 4, 4, 4 4, 4 Since the occurrence of event A affects the probability of event B, the events are dependent. P(A and B) = P(A) P(B A) = 1 1 = 1 4 Hint First circle or highlight the row(s) or column(s) with the given condition(s), that is, event A. Example Using a Table to Find Conditional Probability The table shows the results of a survey that asked students to state the year and season of their birthday Winter Spring a. Find P(Winter 199). Summer SOLUTION P(Winter 199) is the Fall probability of a birthday occurring in the winter, given that it occurred in 199. Since the condition is that it must be in the year 199, focus on the 199 column. winter all seasons : = % b. Find P(199 Spring or Summer). SOLUTION P(199 Spring or Summer) is the probability of a birthday occurring in 199, given that it occurred in the spring or summer. Since the condition is that it must be in the spring or summer, focus on those rows. 199 spring or summer all years : = = % When items are selected and not replaced, the events are dependent. For each successive selection, the total number of outcomes, which are the values in the denominators, decreases by one. 484 Saxon Algebra

50 Example Using a Calculator to Find Conditional Probability A raffle is being held at a school function. A container has the names of the 587 juniors and 618 seniors who are participating on slips of paper. Prizes are being given to the first, second, and third names pulled from the container. Once a student wins a prize, his or her name is not returned to the container. What is the probability that the first two prizes go to juniors and the third prize goes to a senior? Math Reasoning Analyze Calculate P(junior, senior, junior). How does it compare to Example? SOLUTION After the first junior wins, there is one less junior name in the container, which is also one less name overall. After the second junior wins, there is one less name overall again. There are = 105 names in all. P( junior, junior, senior)= The probability is about 1.%. Math Reasoning Estimate Estimate the probability that an envelope was stamped on Thursday given that it was stamped upside down. Example 4 Application: Quality Control At a mailing center, it was found that on Tuesday, 44 out of 66 envelopes were stamped upside down, on Wednesday, 16 out of 51 envelopes were stamped upside down, and on Thursday, out of 598 envelopes were stamped upside down. Find the probability that an envelope was stamped on Wednesday given that it was stamped right-side up. SOLUTION 1. Understand Make a table to organize the information. Subtract to find the number of envelopes stamped right-side up. Tues. Wed. Thurs. Right-side Up Upside Down Plan The condition is that the envelope was stamped right-side up. Find the sum of the envelopes stamped right-side up; this will be the denominator.. Solve Use a calculator = The probability is about 16.%. 4. Check Use estimation to check for reasonableness. From the given information, about 1500 envelopes were stamped and about 50 of those were stamped on Wednesday. The majority of those stamped on Wednesday were stamped right-side up, and = 1 6 = 16 %. Lesson

51 Lesson Practice Suppose the two spinners shown are each spun once. (Ex 1) a. Find the probability that the first spinner lands on 4 and the product 1 of the spinners is less than 10. b. Find the probability that the first 4 spinner lands on a number less than 4 and the sum of the numbers is less than The table shows the results of a survey that 10 th 11 th 1 th asked students from three grade levels to pick Math their favorite subject from those listed. (Ex ) History c. Find P(Math 1). Science d. Find P(10th History). English e. A raffle is being held at a mall. A box (Ex ) has the names of the 7 men and 18 women who are participating on slips of paper. Gift certificates are being given to the first five names pulled from the box. Once a person wins a certificate, his or her name is not returned to the box. What is the probability that the first three certificates go to men and the last two certificates go to women? f. At a factory, it was found that during the morning shift, out of (Ex 4) the 666 boxes of nails that were filled had unacceptable weights, and during the evening shift, 11 out of the 1870 boxes of nails that were filled had unacceptable weights. Find the probability that a box was filled in the evening, given that its weight was found acceptable. Practice Distributed and Integrated 1. Solve s - = -11. Write the solutions in terms of i. (6) Factor completely.. -pax + pax + pa. 0 + x - 1x () () 4. Multi-Step Use x - 6x = 6. a. Explain why the left side of the equation is a perfect square trinomial. (58) b. Factor x - 6x c. Solve the equation for x. 486 Saxon Algebra

52 *5. Error Analysis Find and correct the error a student made in determining the roots (66) and their multiplicity in x - 4x + 4x = 0. x - 4x + 4x = 0 x(x - 4x + 4) = 0 x(x - )(x - ) = 0 The only root is and its multiplicity is. 6. Analyze What does it mean for an event to have the first number in the odds in favor ratio greater than the second number? (55) 7. Geometry Write an expanded expression for the volume of the cube shown. (49) b + 5 cm *8. Housing The table shows estimated housing (68) occupancy for three states for the year 006. Find the probability that a randomly selected house from one of these states is in Nevada. Then find the probability that a randomly selected house from one of these states is in Nevada given that it is vacant. Round each percent to the nearest tenth. Occupied Houses Vacant Houses Mississippi 1,075,51 165,918 Nevada 96,88 18,49 New York 7,088,76 819,18 Simplify (59) (59) (59) (-8) 1. Find the arc length of a sector with a radius of 4 in. and central angle θ = π (6) *1. Probability A bag contains 0 red marbles and 10 white marbles. If three marbles (68) are randomly chosen without replacement, what is the probability of selecting all red or all white marbles? 14. Surveying A surveyor is standing on top of a hill that is 600 meters from a bridge. (46) Her angle of elevation to the top of the bridge is 14 and her angle of depression to the bottom of the bridge is 6. To the nearest tenth of a meter, what is the height of the bridge? *15. Generalize Insert = or to make each statement true. (68) If A and B are independent events, then P(B A) If A and B are dependent events, then P(B A) P(B). P(B). 16. Statistics How do probability and statistics compare and contrast? (60) 17. Multiple Choice Which angle does not have a reference angle of 0º? (56) A -0º B 10º C 570º D -150º 6. Lesson

53 *18. Estimate Use what you know about the unit circle and special angles to estimate (67) Cos -1 (0.48) in degrees. Do not use a calculator. Explain your method. 19. Write Describe how to determine cos 150 by using its reference angle. (6) Simplify (44) 5 (44) (44) Calculate the binomial probability P(1, 00, 0.09) to the nearest hundredth of a N! percent using the formula P(n, N, p) = n!(n - n)! pn (1 - p ) N-n. (Inv 5) *. Graphing Calculator Describe the feasible region for the set of inequalities x 0, (54) x 4, y 0, y + x Error Analysis Find and correct the error a student made in subtracting complex (6) numbers. (7-4i) - ( + i) 5 - i *5. Multiple Choice Which describes all the values of sin (67) A 5π B π 6 6 C π D π -1 1? 6π 6. Estimate Sketch a graph of the equation y = x, where y is the radius of (59) π a sphere and x is the volume of the sphere. Use the graph to estimate the volume of a sphere with a radius of cm. Use the graph to estimate the radius of a sphere with a volume of 14 cm. 7. Volume The volume of a California king size mattress is 1x - 6x - 4x - 84 (61) cubic inches. Factor the polynomial given that x - 8 is a factor. Then find the volume given that x = 0. *8. Plant Nutrition The ph of a soil is given by the function ph = -log [H + ], where (64) [H + ] represents the concentration of hydrogen ions in moles per liter. Most plants require soil with a hydrogen ion concentration between and moles per liter. What ph values correspond to these hydrogen ion concentration values? *9. Travel A driver leaves a parking lot and heads due south at a rate of 40 miles per (65) hour. An hour later, another driver leaves the same parking lot and heads due west at a rate of 60 miles per hour. Approximately how long will it take for the cars to be 00 miles apart? *0. Multi-Step Two bags each contain papers labeled with numbers 1 through 5. One (68) paper is selected from each. a. Explain why selecting a 5 from the first bag and the sum of the numbers is greater than 5 are dependent events. b. Find the probability of both events. 488 Saxon Algebra

54 69 LESSON Simplifying Complex Expressions Warm Up New Concepts Math Language The imaginary axis is sometimes referred to as the i-axis. 1. Vocabulary A number that can be written in the form a + bi where a and b are real numbers is a.. Multiply (x - 5)(4x + ). (6) (19) 5. Simplify Add ( + i) + (6-8i). (40) (6) Complex numbers (a + bi) can be graphed on a complex plane, where the horizontal axis represents the real part, a, and the vertical axis represents the imaginary part, bi. Example 1 Graphing Complex Numbers Graph each number on the complex plane. a. + 4i b. - - i c. -4i d. 1 SOLUTION a. Move units to the right and 4 units up. b. Move units to the left and units down. c. Move 0 units right or left and 4 units down. d. Move 1 unit to the right and 0 units up or down. imaginary 4i + 4i i i - - i -4i -4i real The absolute value of a complex number is its distance from the origin. Unless the point is on the x- or i-axis, the distance is the length of a line segment that forms the hypotenuse of a right triangle and can be found using the Pythagorean Theorem. Absolute Value of a Complex Number imaginary Online Connection a + bi = a + b 4i a + bi i i a + bi b 4 a real Lesson

55 Example Finding the Absolute Value of Complex Numbers Find the absolute value of each number. a. + 4i SOLUTION a + bi = a + b + 4i = + 4 = = 5 = 5 Math Reasoning Verify Use the formula for the absolute value of a complex number to show that -1 is 1. b. - - i SOLUTION a + bi = a + b c. -5i - - i = (-) + (-) = = 1 SOLUTION a + bi = a + b -5i = 0 + (-5) = 5 = 5 When multiplying complex numbers, remember that i = -1. Example Multiplying Complex Numbers Multiply. Write answers in the form a + bi. a. -4i(7 + i) SOLUTION -4i(7 + i) -8i - 1i Distributive Property -8i - 1(-1) i = -1-8i + 1 Simplify. 1-8i Write in standard form. b. (5 + 4i)(6 - i) SOLUTION (5 + 4i)(6 - i) 0-10i + 4i - 8i Use the FOIL method i - 8i Simplify i - 8(-1) i = i + 8 Simplify i Write in standard form. 490 Saxon Algebra

56 Study the powers of i below. Powers of i i 1 = i i 5 = i i 4 = i(1) = i i = -1 i 6 = i i 5 = i(i) = i = -1 i = i i = i(-1) = -i i 7 = i i 6 = i(-1) = -i i 4 = i i = i(-i) = -i = -(-1) = 1 i 8 = i i 7 = i(-i) = -i = -(-1) = 1 Notice that the four simplified values of i, -1, -i, and 1 repeat such that i = i 5, i = i 6, i = i 7, and i 4 = i 8. Any power of i can be found by dividing the exponent of i by four and using the remainder. When the remainder is The expression is equivalent to 0 i 4 = 1 1 i 1 = i i = -1 i = -i Example 4 Simplify each expression. Evaluating Powers of i a. i 0 b. 5i 1 SOLUTION 0 4 = 7 R, SOLUTION 1 4 = R 0, so i 1 = i 4 = 1. so i 0 = i = -1. 5i 1 = 5(1) = 5 c. -i 1 SOLUTION 1 4 = R 1, so i 1 = i 1 = i. -i 1 = -(i) = -i Caution A simplified expression does not have a radical in the denominator. Because i = -1, an expression with i in the denominator is not simplified. To eliminate a radical in a denominator, multiply both the numerator and denominator of the fraction by the conjugate of the denominator. The complex conjugate of a + bi is a - bi. Example 5 Dividing Complex Numbers a. Divide by 6 + i 5. Write the answer in the form a + bi. SOLUTION 6 + i 6 + i i 5 ) 5 ( 6 - i 18 - i 5 Distribute and simplify. 6-5i 18 - i 5 = i Multiply the numerator and the denominator by the complex conjugate of the denominator. Lesson

57 Math Reasoning Analyze Why is -6i the conjugate of 6i? b. Divide 4 + i by 6i. Write the answer in the form a + bi. SOLUTION 4 + i 6i 4 + i 6i ( -6i -6i ) Multiply the numerator and the denominator by the complex conjugate of the denominator. -4i - 6i -6i Distribute. -4i - 6(-1) -6(-1) Simplify. -4i + 6 = i = - i = i Lesson Practice Graph each number on the complex plane. (Ex 1) a. 1 - i b i c. - d. i Find the absolute value of each number. (Ex ) e. 7 - i f i g. 10i Multiply. Write answers in the form a + bi. (Ex ) h. 8i(-9-5i) i. ( - i)(5 + 9i) Simplify each expression. (Ex 4) j. i 18 k. -i 11 l. 1 i1 m. Divide + i by i. Write the answer in the form a + bi. (Ex 5) n. Divide -4 by + i. Write the answer in the form a + bi. (Ex 5) Practice Distributed and Integrated Solve using the quadratic formula. 1. f(x) = x + 8x -. f(x) = x - 10x + 4 (65) (65). Probability A bag holds 5 yellow marbles, 17 red marbles, and 0 green marbles. What is the probability of not choosing a red marble? (55) 4. A savings account earns interest at an annual rate of.8%, compounded quarterly. If the account begins with a principal amount of $,700, what will its value be after 14 years? (57) Solve for x. 5. x 1 7 = 5 6. x = 6 (59) (59) 49 Saxon Algebra

58 *7. Multiple Choice In which quadrant of the complex plane is 5 - i located? (69) A Quadrant I B Quadrant II C Quadrant III D Quadrant IV 8. Write Explain how to use synthetic division to determine if x + 6 is a factor of x + x - 8x - 60 and then perform the division. (61) Simplify (6) (6) 11. Given θ = -90, find the measure of the reference angle. (56) *1. Generalize Show that the product of any two complex conjugates, a + bi and (69) a - bi, is a real number. *1. Evaluate the inverse trigonometric function Sin -1 (- ). Give your answer in both (67) radians and degrees. 14. Why can t the matrix below be used as an encoding matrix for a cipher? (Inv 4) Amusement Parks You get on a ferris wheel ride, and the ferris wheel turns through (56) 170. How much further must the wheel turn before you are at the bottom again? *16. Error Analysis Find and correct the error a student made in finding P(AM Male). (68) P(AM Male) = 16 0 = 4 5 = 80% *17. Graphing Calculator a. Graph y = x + 6x - on a graphing calculator. What are (66) the x-intercepts? b. Find the multiplicity of each root of x + 6x - = 0 by factoring. *18. Formulate Give a rule for finding P(B A). Hint: Consider the rule for P(A and B) (68) for dependent events. 19. Multiple Choice Which equation is equivalent to 10 x = 0.1? (64) A log = x B log 0.1 x = 10 C log = x D log 10 x = 0.1 Male Female AM 16 4 PM 8 1 Lesson 69 49

59 0. Formulate Write a formula for P(A or B or C). Use the diagram to the (60) right to help you. (60) A B C 1. Multi-Step Let f(θ) = Sinθ and g(θ) = Sin -1 θ, where θ is in radians. (67) a. Evaluate f (g ( 1 ) ). Show your steps. b. Evaluate g (f ( π 6 ) ). Show your steps. c. Based on part a, what is the value of f(g(x))? Based on part b, what is the value of g( f (x))?. Error Analysis Examine the work below. Find and explain the error. Then find the (58) correct solution. x - 44x = 11 (x - ) = 11 x - = 11 x = *. Probability A student graphs y = 9x - 9x + and uses a graphing calculator (65) to find the value of the zeros one at a time. What is the probability that the first zero the student finds is negative? Why? *4. Multi-Step Simplify i + i 15. Then graph the number on the complex plane. (69) 5. Planet Rotation Venus is only slightly smaller than Earth; its radius is (6) approximately 757 miles. But it rotates much more slowly; it takes 4 Earth days for Venus to make one complete rotation. If an object is fixed on Venus s equator, how far does it travel in 1 hour due to Venus s rotation? 6. Analyze Describe the value(s) of b x when b = 1 and when b 1. (64) *7. Find the absolute value of - 4i. *8. Find the absolute value of 6 + i. (69) (69) 9. Sports A football player playing in a domed stadium kicks a football straight (6) up into the air. The height of the football after t seconds can be modeled by -16t + 80t. The dome has a height of about 88 feet. The solutions of -16t + 80t = 88 give the number of seconds it will take for the football to reach the top of the dome. Solve the equation. Will the football reach the top of the dome? If so, when? 0. Geometry Find the area of the triangle to the right. (59) 7 () 9cm 11 () 9cm 494 Saxon Algebra

60 70 LESSON Solving Radical Equations Warm Up New Concepts 1. Vocabulary In the expression x, is referred to as the.. Simplify 64. (40) (40). Simplify -15. (40) 4. Solve x + 5x - 0 = 6. (5) 5. Write (x + 6 ) 1 in radical form. (40) A radical equation is an equation that contains at least one radical whose radicand contains a variable. Radical Equations x = 4 x + 5 = x - 1 Not Radical Equations x - 5 = = x - Math Reasoning Analyze Does cubing each side of x = 5 result in extraneous solutions? Explain. To solve a radical equation, first isolate the radical on one side of the equation. Then raise both sides of the equation to the power of the index of the radical. This will clear the radical. It is critical that the solutions of a radical equation be checked because some of the solutions obtained may be extraneous solutions. An extraneous solution is a solution of a derived equation that is not a solution of the original equation. For example, the equation x = 5 has one solution, 5. But when both sides of the equation are squared, the resulting equation x = 5 has two solutions, -5 and 5. The solution -5 is extraneous because it is not a solution of the original equation. Example 1 Solve each equation. a. x - 5 = 14 Solving Equations Containing One Square Root SOLUTION The radical is already isolated on the left side. x - 5 = 14 ( x - 5 ) = 14 x - 5 = 196 x = 01 Square both sides of the equation. Simplify. Add 5 to both sides. Online Connection Check x - 5 = = 14 Lesson

61 b. 6x = 9 SOLUTION First isolate the radical on the left. 6x = 9 6x + 7 = 6 ( 6x + 7 ) = 6 6x + 7 = 6 6x = 9 x = 9 6 Check 6x = 9 6 ( 9 6 ) = 9 Subtract from both sides of the equation. Square both sides. Simplify. Subtract 7 from both sides. Divide both sides by 6. Example Solve each equation. a. 4x = 0 Solving Equations Containing One Cube Root SOLUTION First isolate the radical. 4x = 0 4x + 4 = 6 ( 4x + 4 ) = 6 4x + 4 = 16 4x = 1 Add 6 to each side. Cube both sides to clear the cube root. Simplify. Subtract 4 from each side. x = 5 Divide both sides by 4. Check 4x = 0 4(5) = 0 Math Reasoning Verify Check that -5 is a solution of - x - 8 = Saxon Algebra b. - x - 8 = 14 SOLUTION Isolate the radical by dividing. - x - 8 = 14 x - 8 = -7 Divide both sides by -. ( x - 8 ) = (-7) x - 8 = -4 x = -5 Cube both sides. Simplify. Add 8 to each side.

62 Math Language Indices is the plural of index. Sometimes an equation has two radicals. If their indices are the same and each radical is on a different side of the equation, it is simplest to raise each side of the equation to the power of the index. If the indices on the radicals differ, the solution can be estimated by using a graphing calculator. Graphing calculators are also very useful in checking solutions. Example Solving Equations Containing Two Radicals Solve each equation. Check the solutions graphically. a. 8x - 15 = 4x + 5 SOLUTION 8x - 15 = 4x + 5 ( 8x - 15 ) = ( 4x + 5 ) 8x - 15 = 4x + 5 Square both sides. Simplify. 4x = 0 Isolate x. x = 5 Check Graph Y1 = 8x - 15 and Y = 4x + 5. The intersection point is the solution. Use the intersect command located in the Calculate (nd Trace) menu. The curves intersect at x = 5. Graphing Calculator Tips To type the cube root symbol, press Math and choose it from the items in the Math list. b. x = x - 1 SOLUTION x = x - 1 ( x ) = ( x - 1 ) x = 8(x - 1) Cube both sides. Simplify. x = 8x - 8 Distribute 8. -5x = -8 Isolate x. x = 8 5 = 1.6 Check Graph Y1 = x and Y = x - 1. The curves intersect at x = 1.6. Lesson

63 c. x - -5x - 6 = 0 SOLUTION Move one of the radicals to the right side. Then use a graphing calculator. x - -5x - 6 = 0 x - 6 = -5x Add -5x to each side. Graph Y1 = x - 6 and Y = -5x. The curves intersect at x 4.8. Math Reasoning Analyze How can you see in the first step of 4a that the equation has no solutions? Example 4 Solve each equation. a. x - 4 = -8 SOLUTION x - 4 = -8 Solving Equations with Extraneous Solutions x = -4 ( x ) = (-4) x = 16 Add 4 to each side to isolate the radical. Square both sides. Simplify. Check x - 4 = The solution of 16 is extraneous. The equation has no solutions. b. x + 1 = x + 10 SOLUTION x + 1 = x + 10 (x + 1) = ( x + 10 ) x + x + 1 = x + 10 x - 9 = 0 x = 9 x = ± The radical is isolated on the right. Square both sides. Simplify. Solve the quadratic equation. Check x + 1 = x + 10 x + 1 = x () (-) = 4 - The solution of - is extraneous. The only solution is. 498 Saxon Algebra

64 Example 5 Application: Physics The period of a pendulum, T, is the time it takes for the pendulum to make one back-and-forth swing, and is given by T = π L 9.8, where L is the length of the pendulum in meters. Find the length of a pendulum whose period is seconds. SOLUTION Substitute for T in the formula and solve for L. ( = π L 9.8 L π = 9.8 π ) = ( L 9.8 ) 9 4π = L 9.8 Isolate the variable. Square both sides. Simplify. 9.8 ( 9 4π ) = L Isolate L.. L The length of the pendulum is about. meters. Lesson Practice Solve each equation. a. x + = 1 (Ex 1) (Ex 1) c. x + 1 = 0 (Ex ) (Ex ) b. 11 = x d. 4 x - 4 = 16 e. 9x = 5x + f. - x = 4x + 7 (Ex ) (Ex ) g. x + x = 0 (Ex ) h. -1-4x = x - 11 (Ex 4) (Ex 4) (Ex 5) i. x = 6-5x j. The period of a pendulum, T, is the time it takes for the pendulum to make one back-and-forth swing, and is given by T = π L 9.8, where L is the length of the pendulum in meters. Find the length of a pendulum whose period is 5 seconds. Practice Distributed and Integrated Solve the equations. Write the solutions in the form a + bi. 1. p + 7 = -9p + 4. x + 1 = 9 (6) (6) *. Find the roots and give the multiplicity of each root: x - 7x + 11x - 5 = 0 (66) 4. Let f (x) = x + 1 and g(x) = x -. Find the composite function f (g(x)). (5) Lesson

65 5. Of the 0 people who came into the Italian deli on Friday, 104 bought pizza and 8 used a credit card. Half of the people who bought pizza used a credit card. What is the probability that a customer bought pizza or used a credit card? (60) Solve for x by completing the square. 6. x = 9x x - 6 = -x (58) (58) *8. Lighthouses The number of miles m a person can see on a clear day is (70) approximated by m = h, where h is the person s height in feet above sea level. If a person standing at the top of the Barnegat Lighthouse in New Jersey on a clear day can see a distance of 16 miles, what is the person s height above sea level? 9. Error Analysis The odds in favor of an event are 9 to. To the nearest percent, what is the probability of the event? (55) 10. Multiple Choice What is P(-) if x + is a factor of x + x - 10x + 6? (61) A - B 0 C D Velocity of an Electron The velocity at which an electron orbits the nucleus of an atom (59) is known as the Fermi-Thomas velocity and can be found using the expression Z 17 c, where Z is the number of protons in the nucleus and c is the speed of light. What is the Fermi-Thomas velocity for cobalt, which has 7 protons, in terms of c? Round to the nearest thousandths place. *1. Write What is an extraneous solution? When can it occur? (70) 1. Estimate Use what you know about the unit circle and special angles to estimate (6) tan 4. Do not use a calculator. Explain your method. *14. Graphing Calculator With your calculator in a + bi mode, find the following. (69) a. (64-8i)(6 + 1i) b. -.7i( i) 15. Electrical Engineering A circuit has a current of 5 amps, and a second circuit has a (6) current of (14-9i) amps. Find the sum of the currents. 16. Temporary Employees A department store is planning to hire up to 4 temporary (54) employees for a tent sale. Experienced workers will be paid $0 per hour and inexperienced workers will be paid $15 per hour. The company can pay up to $400 per hour for the temporary employees. Experienced workers produce 1.5 times more profit than inexperienced workers. How many of each type of worker should be hired to maximize the company s profits? *17. Multi-Step The product of a number and its multiplicative inverse is 1. (69) a. State the multiplicative inverse of + i. b. Simplify the multiplicative inverse. c. Verify Show that your answer in part b is the multiplicative inverse by multiplying it by + i. 500 Saxon Algebra

66 Factor each expression completely c z + 0 (61) (61) 0. Generalize How does the Rational Root Theorem make factoring polynomials (66) easier? What must be true about a polynomial before it used? *1. Wheelchair Accessibility A contractor designed the wheelchair ramp shown in (67) the diagram. The ramp must satisfy a building code that specifies 4.7 as the maximum angle of elevation, labeled θ in the diagram. Does the design satisfy the building code? Justify your answer. θ 50 ft 4 ft. Find an equation for the inverse of y = 5x - 0. Determine whether each relation (50) is a function.. Verify A student used factoring to find that the solutions of 4x - 11x - 0 = 0 (65) are -1.5 and 4. Show how to verify these solutions with the Quadratic Formula. 4. Maximize P = 4.5x +.4y for the constraints (54) y 0 x 0 y x y -x Error Analysis A student converted the exponential equation 4 - = 1 to logarithmic (64) 16 form by writing log 4 (-) = 1. What is the error? Write the correct logarithmic 16 equation. 6. Geometry Each side of the regular hexagon shown is 7 cm. Find (5) the length of the diagonal AE. Round your answer to the nearest tenth. (The measure of one interior angle of an n-gon (n - )180 is n.) F A B C E D 7. Write Examine the formula for the probability of two inclusive events. Explain (60) why P(A and B) is subtracted from the formula. P(A or B) = P(A) + P(B) - P(A and B) 8. Multi-Step A clothing store has a sale in which all the shirts are 0% off. On the (5) last day of the sale, all shirts have an additional 5% off. Find a function describing how much a shirt with price p will cost on the last day. Solve the function. *9. Probability A number cube is rolled twice. What is the probability that the number (70) facing up on each roll shows a solution of x - 8 = x - 4? *0. Multiple Choice A bag of grapes contains 9 green grapes and 15 red grapes. If 4 grapes (68) are chosen at random, without replacement, what is the approximate probability that the first two grapes are green and the last two grapes are red? A 4.6% B 5.9% C 7.1% D 1% Lesson

67 INVESTIGATION 7 Collecting Data Exercise Suppose you want to find out what students at your school do for exercise, and then make a graph to display the results. 1. You construct this survey question: What do you do for exercise? Think about some possible responses. How many different responses do you think you would get if you asked 100 students?. Is dance a response you thought of for question 1? Is skateboarding or martial arts a response you thought of?. Is it possible that some students do more than one type of exercise? Online Connection The answers to survey questions are data items, so collecting the answers is a form of collecting data. When constructing a survey question, it is often important to limit the number of possible responses so that you will be able to organize, display, analyze, and interpret the data you collect. Provide enough response choices so that there is a choice for everyone, and make sure the choices don t overlap. That is, construct them so that there is clearly only one response. In the table below, the question about exercise is rephrased and answer choices are supplied, along with numbers of responses. Which of the following best describes your main type of exercise? Math Reasoning Response Analyze Refer to the table. Which choice is needed so that everyone can choose at least one response? What word(s) in the question allows everyone to be able to choose exactly one response? Frequency A. Competitive sports (team and individual) 4 B. Run, jog, or walk (not competitive) 16 C. Swim or bicycle (not competitive) D. Strength training (weights and exercise equipment) E. Other A bar graph and a circle graph are both appropriate ways to display this data. Copy and complete the bar graph and circle graph below Saxon Algebra Competitive Sports 0% % Other 0 Strength Training 5 Swim or Bicycle = Run, Jog, or Walk 80 Exercise 0 Competitive Sports 4 of 80 responses were Competitive. To get the central angle measure for the Competitve sector in the circle graph, multiply 60 by the appropriate decimal. 4 = Exercise Number of Students Hint % % 6%

68 Analyze and Write A survey question is biased if there is a reasonable chance that it could influence a response because of the way it is constructed. Determine whether each question below is biased. Explain your answer. If the question is biased, rewrite it so that it is not biased. 6. Do you agree with most high school students that parents give them too little freedom to make their own decisions? 7. Do you support the unfair policy of requiring community service as a high school requirement? 8. Do you think energy conservation should be a high priority for state and national elected officials? 9. Write a biased survey question. Then rewrite it to remove the bias. A population is a group being studied. Populations are often too large to survey, so a subset of the population called a sample is surveyed. If a sample is not a fair representation of the population, it is biased. If a sample is a fair representation of the population, it is unbiased. When properly chosen in an appropriate situation, any of the following can be an unbiased sample. Random sample Each member of the population has an equal chance of being selected. Systematic sample Members are chosen based on a rule. Stratified sample The population is divided into non-overlapping subgroups, and members are chosen at random from the subgroups. Self-selected sample Members volunteer to be in the sample. Analyze and Write Identify the type of sample and determine whether it is biased. Explain your answer. 10. A manager asks every tenth customer who enters her store whether they like the recent remodeling of the store. 11. A candidate mails a survey about an upcoming election along with a request for campaign contributions, enclosing a return envelope for responses. 1. For a survey about school policies, a student randomly chooses 5 students from each grade. (There are approximately 00 students in each grade.) 1. There are 645 employees at a company. The president assigns the integers 1 through 645, one to each employee, and then randomly chooses 50 integers and surveys the corresponding employees. 14. Every voter in a town is registered as a Democrat, a Republican, or an Independent. For a survey of registered voters, 50 are randomly chosen from each group. 15. Multi-Step Complete parts a d. a. Construct a survey question with 4 or 5 response choices. b. Identify a population. Describe or 4 possible samples of different types. Identify each type and state whether it is biased or unbiased. Investigation 7 50

69 Math Reasoning Analyze A bar graph and circle graph both show relative sizes of categories. Which type better shows how the size of each category compares to the whole? c. Construct a frequency table. Show a reasonable distribution of responses. d. Construct a bar graph and circle graph to show the data. It is useful to describe some data sets with measures of center such as mode, median, and mean, and with measures of spread such as variance and standard deviation. A teacher chose students at random from each of the 10 sophomore homeroom classes in a school and asked To the nearest half hour, what is the average number of hours per school night that you spend on homework? The table below shows the data distribution. Hours Frequency Hint To find the median, order the 0 data values from least to greatest, and then find the average of the 10th and 11th values or more Identify the population and the type of sample. Determine whether the sample is biased. 17. Find the mode(s) and median. 18. Supply the missing terms in the numerator, and then evaluate the 1(0.5) + 4(1.0) + 5(1.5) + + 1(.5) expression to find the mean: x =. Reading Math The symbol for standard deviation is σ, the Greek lower case letter sigma. 0 (x1 - x) + (x - x) + + 1(xn - x) 19. The formula for variance is σ =, where n x, x,, x are the data values and x is the mean. Supply the missing 1 n terms in the numerator, and then evaluate the expression to find the ) + 5(1.5-1(0.5 - x) + 4(1.0 - x x) + + 1(.5 - x) variance: σ = 0 σ. 0. The standard deviation is the square root of the variance: σ = Find the standard deviation to the nearest thousandth. Investigation Practice a. Model Students were asked the survey question, On average, how many hours per week do you or use the Internet? Round to the nearest hour. The frequency table below shows the results. Construct a histogram and a circle graph to show the data. (A histogram is a bar graph in which adjacent bars touch.) Response or more Frequency b. Analyze Suppose the instruction Round to the nearest hour were not included in the survey question in part a. Why might it be difficult for a student to choose a response? c. Analyze Suppose the response choices were 0, 6, 6 9, 9 1, and 1 or more. Why might it be difficult to interpret a histogram showing the results? 504 Saxon Algebra

70 Determine whether each question below is biased. Explain your answer. If the question is biased, rewrite it so that it is not biased. d. How many people live in your household? e. Knowing that teenagers tend to be irresponsible, should people younger than 18 be allowed to drive without an adult in the vehicle? f. Knowing that the current sales tax rate is 5%, do you support a new rate of 6%, beginning next year? g. There are some scientists who believe that recent climate changes are part of a natural long-term climate cycle rather than global warming caused by humans. Do you believe that global warming is a serious problem? Identify the type of sample and determine whether it is biased. Explain your answer. h. A utility company mails a survey to all its customers asking them to rate customer service, enclosing a return envelope for responses. i. A developer wants to build a store. Town council members want to know if town residents approve. They survey all residents that live within a half mile of the proposed site. j. A school principal selects students at random from each class in the school to survey about students study habits. k. The junior class president wants opinions on student parking rules. She uses a list of all 55 juniors, and assigns each junior a different integer from 1 through 55. She uses a random integer generator on a calculator until she gets 0 different integers from 1 through 55. She chooses the 0 corresponding students to survey. Six students were chosen at random from each grade in a high school and asked At what age did you have your first job? The table below shows the data distribution. Age Frequency l. Identify the population and the type of sample. Determine whether the sample is biased. m. Find the mode, median, and mean. n. Find the variance. o. Find the standard deviation. p. Construct your own survey question. Identify the population and the type of sample. Determine whether the sample is biased. Find the mode(s), median, and mean. Find the variance and standard deviation. Investigation 7 505