inequalities involving absolute value. value with both equalities and x - 5 = 4 3 x + 7 = 14 inequalities. -5 x > 20 2 x + 1 < 6

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1 *1. Students solve equations and 1. Solve equations involving absolute Solve for x. inequalities involving absolute value. value with both equalities and x - 5 = 4 3 x + 7 = 14 inequalities. -5 x > 20 2 x + 1 < 6 Express the solution using interval notation. ( FW) 2x - 3 > 4 Sketch the interval in the real number line that is the solution for: ( FW) x - 3 < 5 2 *2. Students solve systems of linear equations and inequalities (in two or three variables) simultaneously, by substitution, graphically, or with matrices. 1. Demonstrate methods of solving systems of linear equations (inequalities). Substitution Linear combination Graphically Using matrices Solve for x and y algebraically for each system and verify the solutions graphically or with matrices. a) y < 5x - 4 2x + 3y > 27 b) -x + y = 3 2x - y = -5 c) 2x + z = 7 x + y + z = 0 2x + 3y - 2z = -8 1

2 2

3 Solve the system of linear equations: ( FW) x + 2y = 0 x + z = -1 y - z = 2 Draw the region in the plane that is the solution set for the inequality. ( FW) (x - 1) (x + 2y) > 0 *3. Students are adept at operations on polynomials, including long division. 1. Use operations with polynomials. Addition Subtraction Multiplication Division Simplify the following completely: (3x 2 + 7x - 12) + (5x 2-8x + 4) ( 5x 2 + 7x + 2) - ( 4x 2-13x - 9) (2x - 7)(3x 2 + 7x -4) x x + 1 Divide x 4-3 x 2 + 3x by x Use this to write x 4-3x 2 + 3x in the x form: polynomial + linear polynomial ( FW) x

4 *4. Students factor polynomials 1. Factor polynomials. representing the difference of squares, Difference of squares x 2-25 perfect square trinomials, and the sum and difference of two cubes. Perfect square trinomials x 2 + 2xy + y2 Sum of two cubes Difference of two cubes x 3 + y 3 8x 3-27y 3 *5. Students demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane. 1. Plot complex numbers as points in a plane. 4 Simplify: ( FW) x 3 - y 3 x 2 - y 2 Express without any square roots in the numerator: ( FW) Graph each complex number in the complex plane. imaginary a) 3 + 2i b) - 5i c) 4.5 x + y x - y 2 real Give the absolute value of the complex number: i

5 *6. Students add, subtract, multiply, and divide complex numbers. *7. Students add, subtract, multiply, divide, reduce and evaluate rational expressions with monomial and polynomial denominators, and simplify complicated fractions including fractions with negative exponents in the denominator. 1. Use operations with complex numbers. Add Subtract Multiply Divide 1. Simplify and evaluate rational expressions. Locate all complex solutions to z in the complex plane. ( FW) Simplify the following completely: (4-5i) + (7 - i) ( 4-2i) - (9 + 5i) (4 + 3i)(4-3i) 7-2i 2 + 1i Write in the form a + bi, where i is a square root of -1: ( FW) (3-2i) i Simplify the following completely: 6-3 2x x - 7 4b 5b - 1, 5x - 3 5x - 3 5a 4 b 3 x 20a 2 b 3, x + 3 x - 2 4a -5 75ab 2 x -2 x + 6 Simplify: (x 2 - x) 2 ( FW) x(x - 1) -2 (x 2 + 3x - 4) 5

6 1. Demonstrate methods of solving quadratic equations. Factoring *8. Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system. Completing the square Solve by factoring y =2x 2-5x -3 Solve by completing the square y = 2x 2 + 5x + 1 Quadratic formula Solve by using the quadratic formula. y = x 2 + 3x - 3, y = x 2 + x + 10 Word problem What is the maximum area of a pillow that has a perimeter of 36 inches? In the figure shown below, the area between the two squares is 11 square inches. The sum of the perimeters of the two squares is 44 inches. Find the length of a side of the larger square. (ICAS) ( FW) Find all solutions to the equation. ( FW) x 2 + 5x + 8 = 0 6

7 1. Determine how the graph of a Find the vertex of y = 10( x - 2) quadratic function changes as a, b, and c vary in an equation. Graph y = -2 (x - 3) *9. Students demonstrate and explain the effect changing a coefficient has on the graph of quadratic functions. That is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b)2 + c. The function f(x) = (x - b) 2 + c is graphed below. Use this information to identify the constants b and c. ( FW) (2,1) y = f (x) x *10. Students graph quadratic functions and determine the maxima, minima, and zeros of the function. 1. Graph quadratic functions and determine their maxima and minima and zeros. For the function h(x) = 3x 2 + 6x - 24 a) Find the vertex b) Find the line of symmetry c) Fine the maxima/minima point d) the zeros Find a quadratic function of x that has zeroes at x = -1 and x = 2. Find a cubic equation of x that has zeroes at x = -1 and x = 2 and nowhere else. (ICAS) ( FW) 7

8 Graph the function f(x) = 2(x + 3) 2-4 and determine the minimum value for the function. ( FW) Find the vertex for the graph of f(x) = 3x 2-12x + 4. ( FW) *11. Students prove simple laws of logarithms. *1. Students understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents. 2. Students judge the validity of an argument based on whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step. Solve each equation. l0g 5 x = 3 x = log (e 4x ) = Determine if the following work is correct or not and state why. logx + log 3 = log 12 log(x + 3) = l0g12 x + 3 = 12 x = 12 Solve for x and explain each step: ( FW) log 3 (x + 1) - log 3 x = 1 log 7 = log b x b 8

9 1. Use the laws of exponents and exponential functions. *12. Students know the laws of exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay. Sharon is investing $1000 in a certificate of deposit that pays 6% interest each year. If she keeps the money in the account, how much will she have after 8 years? You have 10 grams of radioactive isotope gold-202. After 1 minute there are only 8 grams left. How long will it take until there are 5 grams left? The number of bacteria in a colony was growing exponentially. At 1 p.m. yesterday the number of bacteria was 100 and at 3 p.m. yesterday it was How many bacteria were there in the colony at 6 p.m. yesterday? (TIMSS) ( FW) Scientists have observed that living matter contains, in addition to carbon, C12, a fixed percentage of a radioactive isotope of carbon, C14. When the living material dies, the amount of C12 present remains constant, but the amount of C14 9

10 decreases exponentially with a half life of 5,550 years. In 1965, the charcoal from cooking pits found at a site in Newfoundland used by Vikings was analyzed and the percentage of C14 remaining was found to be 88.6%. What was the approximate date of this Viking settlement? (ICAS) ( FW) 13. Students use the definition of logarithms and the product formula for logs to translate between logarithms in any bases. 1. Translate between logarithms in any base. Express each of the following quantities in terms of base ten logarithms. log 2 5 log 5 10 ln3 log b 2 log 2 b Calculate the following logarithms to three decimal places. log 5 14 log 3 1/2 Simplify to find exact numerical values for: log (b 2 ) b 3log b 2-log b 5 b 10

11 1. Simplify logarithmic expressions and approximate their values. 14. Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and identify their approximate values. Give log and log3 0.48, find the approximate value of the following: log6 log8 log(0.75) log2/3 Solve the following equations: 3 x+2 = 81 2logx = log144 Find the largest integer that is less than: ( FW) log 10 (1,256) log 10 (.029) Write as a single logarithm: ( FW) log 3 7 log 3 5 *15. Students determine if a specific algebraic statement involving rational expressions, radical expressions, logarithmic or exponential functions, is sometimes true, always true, or never true. 1. Determine if expressions are true. Determine whether each statement is true for: a) all real numbers. b) some real numbers. c) no real numbers 1/x + 1/2 = x + 2, log (2x) = log(x 2 ) 2x 11

12 a + 3= a = 3 5 x+2 = 25(5 x ) Is the following true for all real numbers x, for some real numbers, or for no real numbers x? ( FW) (1 - x 2 ) x 1 + x = 16. Students demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it. 1. Graph conic sections and determine how the graphs are affected when coefficients are changed. Graph the following equation. x 2 + y 2 = 25 Write and equation for the circle in standard form. Then state the center and radius. x 2-6x + y 2 = 0 Graph the ellipse and give the coordinate of its foci,. Find its eccentricity, and express it as a decimal rounded to the nearest hundredth. 25x 2 + 9y 2 =

13 Graph the hyperbola and give the coordinates of its foci and the equations for its asymptotes. (y + 2) 2 - (x - 3) 2 = If xy = 1 and x is greater than 0, which of the following statements is true? ( FW) a) When x is greater than 1, y is negative. b) When x is greater than 1, y is greater than 1. c) When x is less than 1, y is less than 1. d) As x increases, y increases. e) As x increases, y decreases (TIMSS) ( FW) 17. Given a quadratic equation of the form ax 2 + by 2 + cx + dy + e = 0, students can use the method of completing the square to put the equation into standard form and can recognize whether its graph is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation. 1. Use quadratic equation to recognize its graph. 13 Complete the square for the following equations. Then state if the conic is a circle, an ellipse, a parabola, or a hyperbola. Then give the center and graph the conic.

14 x 2 - y - 8x + 16 = 2 3x 2 + 4y 2-6x + 16y = -7 16x 2-9y 2-32x + 90y = 353 x 2 + 2x + y 2-6y + 11 =9 Does the origin lie inside of, outside of, or on the geometric figure whose equation is x 2 + y 2-10x + 10y - 1 = 0? Explain your reasoning. (ICAS) ( FW) Write the conic section whose equation is given by 4x 2-8x-y 2 + 4y = 4 in standard form to determine whether it is a parabola, hyperbola, or ellipse. ( FW) *18. Students use fundamental counting principles to compute combinations and permutations. 1. Use counting principle. During your turn in a dice game, you rolled two dice. a) How many ways can you get a sum of 5? b) How many ways can you get a sum of 10? c) How many ways can you get a sum of 5 and 10? 14

15 At Romano s Pizzeria, you can order pizza with thick or thin crust and with any combination of the five toppings they offer. a) How many types of crust are there? b) How many choices of toppings are there? c) How many different types of one-topping pizzas can be ordered? An examination consists of 13 questions. A student must answer only one of the first two questions and only nine of the remaining ones. How many choices of questions does the student have? (TIMSS, adapted) ( FW) *19. Students use combinations and permutations to compute probabilities. 1. Compute probabilities. Suppose you draw 2 cards at random from a deck of 52 playing cards. What is the probability that both will be black? Evaluate the following expressions: 8! 8P 6 8C 6 7!/3! 15

16 Six people with different last names line up randomly. What is the probability they are lined up in alphabetical order? A lottery will be held to determine which three members of a club will attend the state convention. This club has 12 members, 5 of whom are women. What is the probability that none of the representatives of the club will be women? *20. Students know the Binomial Theorem and use it to expand binomial expressions which are raised to positive integer powers. 1. Use the Binomial Theorem to expand binomial expressions which are raised to positive powers. Write the first three terms of (a + b) 9 Write an informal description of the following words: a) term b) power c) coefficient d) exponent Answer the following question for the binomial power: (x + 4) 4. a) How many terms are in its expansion? b) Write the coefficients of each term using combination notation. c) Write the complete expansion in simplest form. 16

17 Determine the middle term in the binomial expansion of (ICAS) ( FW) 21. Students apply the method of mathematical induction to prove general statements about the positive integers. 22. Students find the general term and the sums of arithmetic series and both finite and infinite geometric series. 1. Use mathematical induction to prove statements about the positive integers. 1. Fine the general term and the sums of arithmetic series. x - ( 2 x) 10 Use mathematical induction to prove each formula is valid for all positive integral values of n (2n -1) = n n = n(n + 1) n-1 = 2 n - 1 Use mathematical induction to show that n(n + 1) n = 2. ( FW) Put each of the following series or partial sums in sigma notation and find the sum the first 15 terms

18 Find the sum of the following infinite series: ( FW) *23. Students derive the summation formulas for arithmetic series and both finite and infinite geometric series. 24. Students solve problems involving functional concepts such as composition, inverse, and arithmetic operations on functions. 1. Derive the summation formulas for arithmetic series. 1. Use arithmetic inverse and composition concepts on functions ` Show that the nth term of an arithmetic sequence with first term a 1 and common difference d is given by a n = a 1 + (n - 1) d. Given that s 6 = s 6 = s 6 = s 6 (1-6) = s 6 = then derive s n using the same logical steps. Given that f(x) = 2x + 6 and g(x) = 3x - 2; find the following: f + g(x) f - g(x) f x g(x) f g(x) f g(-3) 18

19 Find f -1 (x) when f(x) = x 2-6. Then compute f -1 (-2) Which of the following functions are their own inverse functions? Use at least two different methods to answer this, and explain your methods. (ICAS) ( FW) f(x) = 2 x g(x) = x h(x) = lnx lnx j(x) = 3 x x Students use properties from number systems to justify steps in combining and simplifying functions. 1. Use properties from number systems to justify steps in combining and simplifying functions. Sketch a graph of a function g that satisfies the following conditions: g doesn t have an inverse function, g(x) <x for all x, and g(2) >0. ( FW) Identify the field axiom illustrated below. f(x) + g(x) = g(x) + f(x) f(x) +{g(x) + h(x)} = { f(x) + g(x) + h(x) f(x) { g(x) + h (x) } = f(x)h(x) + g(x)h(x) Let b(x) = x + 3 and c(x) = 4x, show that b(-2) + c(-2) = (b+c)(-2) 19

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