Math Analysis Chapter 2 Notes: Polynomial and Rational Functions

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1 Math Analysis Chapter Notes: Polynomial and Rational Functions Day 14: Section -1 Complex Numbers; Sections - Quadratic Functions -1: Complex Numbers After completing section -1 you should be able to do the following: 1. Add and subtract complex numbers. Multiply complex numbers 3. Divide complex numbers 4. Perform Operations with square roots of negative numbers 5. Solve quadratic equations with complex imaginary solutions In order to take a square root of a negative number mathematicians invented an expanded system of numbers called complex numbers. The Imaginary Unit i The imaginary unit is defined as i 1, where i 1 Complex Numbers The set of all numbers in the form a + bi with real numbers a and b, and i, the imaginary unit are called the set of complex numbers. The real number a is called the real part and the real number b is called the imaginary part of the complex number a + bi. In order to add complex numbers you must add like terms. Practice: In 1-, Add or subtract as indicated: 1. (5 i) + (3 + 3i). ( + 6i) (1 i) In order to multiply complex numbers it is done using the distributive property and the FOIL method. After completing multiplication, we replace any occurrences of i with 1. Practice: In 1-, Find the products: 1. 7i( 9i). (5 + 4i)(6 7i) Math Analysis Notes prepared by Mrs. Atkinson 1

2 Conjugate of a Complex Number The complex conjugate of the number a + bi is a bi, and the complex conjugate of a bi is a + bi. The multiplication of complex conjugates gives a real number: (a + bi)(a bi) a + b (a bi)(a + bi) a + b In order to divide complex numbers is really a process of multiplying by the complex conjugate of the denominator. Remember that you want to keep the expression the same value so you must multiply by 1, so what ever you do to the bottom of a fraction you must do to the top of the fraction. Practice: In 1-, Divide and express the result in standard form: i 4 i i 4i Roots of Negative Numbers Principal Square Root of a Negative Number For any positive real number b, the principal square root of the negative number b is defined by: b i b When performing operations with square roots of negative numbers, begin by expressin all square roots in terms of i. The perform the indicated operations. Practice: In 1-4, Perform the indicated operations and write the result in standard form ( 1+ 5) Math Analysis Notes prepared by Mrs. Atkinson

3 Quadratic Equations with Complex Imaginary Solutions A quadratic equation can be expressed in the general form ax + bx + c 0 and can be solved using the b± b 4ac quadratic formula: x a When solving quadratic equations that have complex imaginary solutions you must express the final answer in standard form a + bi. Practice: Solve using the quadratic formula: x x Quadratic Functions After completing section - you should be able to do the following: 1. Use transformations of the common function y x to determine the characteristics of parabolas.. Graph parabolas (quadratic equations) 3. Determine quadratic function s minimum or maximum value Graph Quadratic Functions in Standard Form The standard Form of a Quadratic Function The quadratic function f ( x) a( x h) + k is in standard form. The graph of f is a parabola whose vertex is at the point (h, k). The parabola is symmetric with respect to the line x h. If a > 0, the parabola opens upward; if a < 0, the parabola opens downward. Steps for changing from general form to standard form of a quadratic function. 1. Move constant term to other side of equal sign.. Factor out a if it is not Complete the square, remember what you add to one side you must add to the other. 4. Factor the x side to a square of a binomial. 5. Resolve for y and the quadratic should now be in standard form: y a(x h) + k Practice: in 1-, Rewrite the quadratic into standard form: 1. f(x) x 6x f(x) 3x + 1x 8 Math Analysis Notes prepared by Mrs. Atkinson 3

4 Graphing Quadratic Functions with Equations in Standard Form 1. Determine whether the parabola opens upward or downward. If a > 0, it opens upward. If a < 0, it opens downward.. Determine the vertex of the parabola. The vertex (h, k). 3. Find any x-intercepts by solving f(x) 0. The function s real zeros are the x-intercepts. 4. Find the y-intercepts by evaluating f(0). 5. Plot the intercepts, the vertex and additional points as necessary. Connect these points with a smooth curve that is shaped line a u. called a parabola. Practice: Graph the quadratic function f(x) x + x + 3. Find the vertex and intercepts to sketch the graph. Give the equation of the parabola s axis of symmetry. Use the graph to determine the function s domain and range (use interval notation). Minimum and Maximum Values of Quadratic Functions. If a parabola opens downward a < 0, then the quadratic function has a maximum at the vertex (h, k). If a parabola opens upward a > 0, then the quadratic functions has a minimum at the vertex (h, k). Practice: In 1-, (a) Determine without graphing, whether the function has a minimum value or a maximum value. (b) Find the order pair were the minimum value or maximum values occur. (c) Identify the function s domain and its range without graphing. 1. f(x) 3x 1x 1. f(x) 4x + 8x 3 Math Analysis Notes prepared by Mrs. Atkinson 4

5 Day 15: Section -3 Polynomial Functions and Their Graphs After completing section -3 you should be able to do the following: 1. Identify polynomial functions and state their degree.. Determine left and right side end behavior 3. Use factoring to find zeros of polynomial functions 4. Identify zeros and their multiplicities 5. Use the Intermediate Value Theorem 6. Understand the relationship between degree and turning points. 7. Graph polynomial functions Definition of a Polynomial Function A polynomial function is a function of the form: 1 ( ) n n n f x anx + an 1x + an x a1x+ a0 The exponents are all whole numbers, and the coefficients are all real numbers. Examples of Polynomials: Degree Type Standard Form 0 Constant f ( x) a0 example: f(x) 10 1 Linear f ( x) a1x+ a0 example: f(x) x + 5 Quadratic f ( x) ax + a1x+ a0 example: f( x) x x Cubic 3 f ( x) a x + a x + a x+ a example: f(x) 5x 3 3x Quartic 4 3 f ( x) a x + a x + a x + a x+ a example: x A polynomial may not have x in the denominator of the expression, also all exponents on the variable must be positive whole numbers. Practice: 1-4, Determine which functions are polynomial functions. For those that are, identify the degree. 1. hx ( ) 7x + x +. x 3 1 f( x) x gx ( ) x x + 4. kx ( ) 5 x x Math Analysis Notes prepared by Mrs. Atkinson 5

6 Graphs of Polynomials are Smooth and Continuous Graphs Polynomial functions of degree or higher have graphs that are smooth and continuous. By smooth, we mean that the graphs contain only rounded curves with no sharp corners. By continuous, we mean that the graphs have no breaks and can be drawn without lifting your pencil from the rectangular coordinate system. Smooth rounded corner Graphs of Polynomials Functions Smooth rounded corner Not Graphs of Polynomials Functions Sharp Corner Smooth rounded corner Smooth rounded corner Smooth rounded corner Discontinuous: a break in the graph Sharp Corner End Behavior of Graphs of Polynomial Functions Odd-degree polynomial functions have graphs with opposite behavior at each end. Even-degree polynomial functions have graphs with the same behavior at each end. The Leading Coefficient Test The coefficient on the term with the largest exponent on the variable is called the leading coefficient. If the largest exponent is an odd number: If the leading coefficient is positive, the graph falls to the left and rises to the right. Rises Left If the leading coefficient is negative, the graph rises to the left and falls to the right. Rises Right Falls Left Falls Right Odd Degree; positive leading coefficient Odd Degree; negative leading coefficient Math Analysis Notes prepared by Mrs. Atkinson 6

7 The Leading Coefficient Test (continued) The coefficient on the term with the largest exponent on the variable is called the leading coefficient. If the largest exponent is an even number: If the leading coefficient is positive, the graph rises to the left and rises to the right. If the leading coefficient is negative, the graph falls to the left and falls to the right. Falls Right Rises Left Rises Right Falls Left Even Degree; positive leading coefficient Even Degree; negative leading coefficient Practice: In 1-4, Use the Leading Coefficient Test to determine the end behavior of the graph of each function: 1. f(x) 5x + x 3x 3. f(x) x 4 + x 3 + 3x + 5x 0 3. f(x) 7x 7 4. f(x) 5x 5 3x 6 Zeros of Polynomial Functions A zero of a polynomial function is found by setting the functions equal to zero and solving the resulting equation. Zeros are also called solutions and roots. If a zero occurs at the same x-value an even number of times the graph of the function at that point touches the x-axis and turns around. If a zero occurs at the same x-value an odd number of times the graph of the function at that point crosses the x-axis. Math Analysis Notes prepared by Mrs. Atkinson 7

8 Zero occurs an odd number of times at x 3, because it crosses the x-axis. Zero occurs an odd number of times at x 4, because it crosses the x-axis. Zero occurs an even number of times at x 1, because it turns around at that point. Practice: In 1-4, Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around, at each zero. 1. f(x) x(x 4) (x + 3) 3. f(x) 4(x 3)(x + ) 3. f(x) x 3 x + x 4. f(x) x 3 + 7x 4x 8 The Intermediate Value Theorem for Polynomials Let f be a polynomial function with real coefficients. If f(a) and f(b) have opposite signs, then there is at least one value of c between a and b for which f(c) 0. Therefore, the equation f(x) 0 has at least one real root between a and b. Math Analysis Notes prepared by Mrs. Atkinson 8

9 Practice: In 1-, Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. 1. f(x) x 3 x 1; between 1 and. f(x) x 3 + x x + 1 between 3 and Graphing a Polynomial Function. 1. Use the Leading Coefficient Test to determine the graph s end behavior.. Find the x-intercepts by setting f(x) 0 and solving the resulting polynomial equation. If the same zero occurs an even number of times, the graph touches the x-axis at that x-value and turns around. If the same zero occurs an odd number of times, the graph crosses the x-axis at that x-value. 3. Find the y-intercept by computing f(0). 4. Use symmetry, if applicable, to help draw the graph. y-axis of symmetry: f( x) f(x) (even function) Origin symmetry: f( x) f(x) (odd function) 5. Use the fact that the maximum number of turning points of the graphs is the degree minus 1 to check whether it is drawn correctly. Practice: In 1-, (a) Use the Leading Coefficient Test to determine the graph s end behavior. (b) Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. (c) Find the y-intercept. (d) Determine whether the graph has y-axis of symmetry, origin symmetry, or neither. (e) If necessary, find a few additional points and graph the function. 1. f(x) x x. f(x) x (x 1) 3 (x + ) Math Analysis Notes prepared by Mrs. Atkinson 9

10 Day 16: Section -4 Dividing Polynomials; Remainder and Factor Theorems; Section -5 Zeros of Polynomial Functions. After completing section -4 you should be able to do the following: 1. Use long division to divide polynomials. Use synthetic division to divide polynomials 3. Evaluate a polynomial using the Remainder Theorem Long Division of Polynomials 1. Arrange the terms of both the dividend and the divisor in descending powers of the variable.. Divide the first term in the dividend by the first term in the divisor. The result is the first term of the quotient. 3. Multiply every term in the divisor by the first term in the quotient. Write the resulting product beneath the dividend with like terms lined up. 4. Subtract the product for the dividend. 5. Bring down the next term in the original dividend and write it next to the remainder to form a new dividend. 6. Use this new expression as the dividend and repeat this process until the remainder can no longer be divided. This will occur when the degree of the remainder (the highest exponent on a variable in the remainder) is less than the degree of the divisor. Practice In 1-, divide using long division. State the quotient, q(x) and the remainder, r(x). 1. (1x + x 4) (3x ). x 8x + x + x 3 x Math Analysis Notes prepared by Mrs. Atkinson 10

11 Synthetic Division: Use only when divisor is a linear expression. To divide a polynomial by x c Example x x x x Arrange the polynomial in descending powers, 1. 6x 5 + 0x 4 x 3 + 0x 3x + 1 with a 0 coefficient for any missing term.. Write c for the divisor, x c. To the right, Write the coefficients of the dividend. 3. Write the leading coefficient of the dividend on the bottom row. Bring down 6 4. Multiply c (in this case ) times the value just written on the bottom row. Write the product 1 in the next column in the second row. 6 6 Multiply by : Add the values in this new column, writing the sum in the bottom row. 1 Add 1 6. Repeat this process of multiplications and additions until all columns are filled in. 1 4 Add 6 1 Multiply by : Use the numbers in the last row to write the Quotient, plus the remainder above the divisor. The degree of the first term of the quotient is one less than the degree of the first term of the dividend. The final value in this row is the remainder Written Form The last row of the synthetic division 6x + 1x + x + 44x+ 85+ x 5 3 x 6x x 3x Math Analysis Notes prepared by Mrs. Atkinson 11

12 Practice: In 1-, divide using synthetic division. 1. (x 5x 5x 3 + x 4 ) (5 + x) x + x 10x + 1 x + The Remainder Theorem If the polynomial f(x) is divided by x c, then the remainder is f(c). Practice: In 1-, use synthetic division and the Remainder Theorem to find the indicated function value. 1. f(x) x 3 7x + 5x 6; f(3). f(x) 6x x 3 + 5x + x + 1; f 3 Zeros of Polynomial Functions After completing section -5 you should be able to do the following: 1. Use the Rational Zero Theorem to find possible rational zeros.. Find zeros of a polynomial function 3. Solve polynomial equations 4. Use the linear Factorization Theorem to find polynomials with given zeros. The Rational Zero Theorem If a polynomial is written in descending order of the variable then a list of possible zeros can be made from: Factors of the constant term Possible rational zeros Factors of the leading coefficient Practice: In 1-, List all possible rational zeros of: 1. f(x) 1x 4 + 4x 5 3 x. f(x) x 3 + x 6 Math Analysis Notes prepared by Mrs. Atkinson 1

13 Finding the zeros of the Polynomial Function. 1. Begin by listing all possible rational zeros.. Use division to show that the remainder is zero. Remember that The Remainder Theorem states the remainder is the value of the function. Practice: Find all the zeros of f(x) x 3 + x 5x Properties of a Polynomial Equation. 1. Counting the number of roots, zeros, or solutions separately, it should equal the degree (highest exponent on the variable) of the polynomial.. If the polynomial has one complex imaginary root a + bi then it will also have it will also have the conjugate complex imaginary root a bi as a root also. Complex imaginary roots, if they exist, occur in conjugate pairs. Practice: Solve: x 4 6x 3 + x 30x The Linear Factorization Theorem n n 1 n If f ( x) a x + a x + a x a x+ a where n 1 and a n 0, then n n 1 n 1 0 ( )( ) ( ) f ( x) an x c1 x c... x cn Where c 1, c,.c n are the zeros of the function. In words, an nth degree polynomial can be expressed as the product of a non zero constant (a n ) and n linear factors. Math Analysis Notes prepared by Mrs. Atkinson 13

14 Practice: Find an nth degree polynomial function with real coefficients satisfying the given conditions n 3; 4 and i are zeros; f( 1) 50. n 4; 4,, and + 3i are zeros; f(1)100 3 Math Analysis Notes prepared by Mrs. Atkinson 14

15 Day 17: Section -6 Rational Functions and Their Graphs After completing section -6 you should be able to do the following: 1. Find the domain of rational functions. Identify vertical asymptotes 3. Identify horizontal asymptotes 4. Identify slant asymptotes 5. Graph rational functions Finding the Domain of a Rational Function. Remember there are two things that restrict the domain and they are radical signs on variables and variables in the denominator of fractions. 1. If you have even indexed radicals set what is under the radical greater than or equal to zero and solve the resulting inequality for the domain.. If you have variables in the denominator set the denominator not equal to zero and sove the resulting inequality for the domain. Practice: in 1-3, Find the domain of each rational function. Write answer in interval notation. x 5 x x f( x). gx ( ) 3. hx ( ) x 5 x 4 x + 16 Vertical Asymptotes of Rational Functions A vertical asymptote is a vertical line in the graph of the rational function that the graph will never intersect. The graph of a rational function may have no vertical asymptote, one vertical asymptote, or several vertical asymptote. We will use dashed lines to show asymptotes. Graph the common function f( x) 1. x Step 1 State Domain The value of the domain that is restricted is the value of the asymptote(s). If the domain is all real numbers then the graph of the rational function has no vertical asymptote. Step Graph the vertical asymptote as a dashed line. Step 3 Choose 3 x-values smaller and larger than each vertical asymptote Step 4 Connect points with smooth curve, make sure you do not intersect the vertical asymptote(s). Math Analysis Notes prepared by Mrs. Atkinson 15

16 Rational Function Rational Function are quotients of polynomial functions: p( x) f( x). qx ( ) Locating Vertical Asymptotes. p( x) If f( x) is a rational function in which p(x) and q(x) have no factors and a is a zero of q(x), the qx ( ) denominator, then x a is a vertical asymptote of the graph of f. The vertical asymptote is related to the domain. A vertical asymptote will exist at each restriction on the domain of the rational function as long as the factor that produced the restriction is not a common factor with the numerator. Practice: In 1-3, Find the vertical asymptotes, if any, of the graph of each rational function: x x 1 x 1 1. f( x). f( x) 3. f( x) x 1 x 1 x + 1 Horizontal Asymptotes of Rational Functions If a function has a value that x that is restricted in it s domain, the rational function may be a y-value the that is also restricted. If a rational function has a restriction in it s y-value we call that y-value a horizontal asymptote at y b, where b equals the y-value that is restricted. Rules for finding Horizontal Asymptotes: p( x) If f( x) is the rational function then: qx ( ) 1. The H.A. is y 0 if the degree of q(x) is larger than p(x).. The H.A. is the ratio of the leading coefficients if the degree of q(x) is equal to the degree of p(x). 3. There is no H.A. if the degree of p(x) is larger than q(x). Practice: Find the horizontal asymptote, if any, of the graph of each rational function: 3 9x 9x 9x 1. f( x). f( x) 3. f( x) 3x + 1 3x + 1 3x + 1 Math Analysis Notes prepared by Mrs. Atkinson 16

17 Graphing Rational Functions 1. Find vertical asymptote(s) and draw as a dashed vertical line on graph. Find horizontal asymptote and draw as a dashed horizontal line on graph 3. Choose 3 points smaller and larger than the for each vertical asymptote. 4. Plot points from step 3 connecting with a smooth curve making sure not to intersect the vertical or horizontal asymptotes. Practice: In 1-4, graph each rational function 1. f( x) 1 1 x +. 3x f( x) x Math Analysis Notes prepared by Mrs. Atkinson 17

18 3. f( x) x x 9 Slant Asymptotes. We have seen that if the degree of the numerator is larger than the denominator then the rational function does not have a horizontal asymptote. Over it may still have y-values that are restricted. The graph of a rational function has a slant asymptote if the degree of the numerator is one more than the degree of the denominator. The slant asymptote is the equation y the quotient of the division of the rational function. x 5x+ 7 Practice: Find the slant asymptote of f( x) x Practice: Graph the rational function: x f( x) 1 x Math Analysis Notes prepared by Mrs. Atkinson 18

19 Day 19: Section -7 Polynomial and Rational Inequalities After completing section -7 you should be able to do the following: 1. Solve polynomial inequalities. Solve rational inequalities Procedure for Solving Polynomial Inequalities 1. Express the inequality in the form: f( x) > 0 or f( x) < 0 where f is a polynomial function.. Solve the equation f(x) 0. The solutions are boundary points. 3. Locate these boundary points on the number line, thereby dividing the number line into intervals. 4. Choose one representative number, called a test value, within each interval and evaluate f at that number. If the value of f is positive, then f(x) > 0 for all numbers x, in the interval. If the value of f is negative, then f(x) < 0 for all numbers x, in the interval. 5. Write the solution set, selecting the interval or intervals that satisfy the given inequality. Practice 1-4: Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. 1. x 5x + 4 > 0. 5x 3x 3. (x 1)(x )(x + 3) > 0 4. x 4 + x 3 + 4x + 4x 0 Math Analysis Notes prepared by Mrs. Atkinson 19

20 Solving Rational Inequalities Procedure for Solving Rational Inequalities 1. Express the inequality so that one side is zero and the other side is a single quotient, if necessary.. Set the numerator and the denominator of f equal to zero. These real solutions are the boundary points 3. Locate the boundary points on a number line and separate the line into intervals 4. Choose one test value within each interval and evaluate f at that number. 5. Write the solutions set, selecting the interval or intervals that satisfy the given inequality. Practice: In 1-4, Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. x + 5 x + x 6 1. > 0. 0 x x x < x + x 4. 0 x + Math Analysis Notes prepared by Mrs. Atkinson 0

21 Day 0: Section -8 Modeling using Variation After completing section -8 you should be able to do the following: 1. Solve direct variation problems. Solve inverse variation problems 3. Solve combined variation problems 4. Solve problems involving joint variation Direct Variation If a situation is described by an equation in the form: y kx, where k is a nonzero constant, we say the y varies directly as x. The number k is called the constant of variation. Procedure for Solving Variation Problems 1. Write an equation that describes the given English statement.. Substitute the given pair of values into the equation in step 1 and solve for k, the constant of variation 3. Substitute the value of k into the equation in step Use the equation from step 3 to answer the problem s question. Practice: 1. The number of gallons of water, W, used when taking a shower varies directly as the time, t, in minutes, in the shower. A shower lasting 5 minutes uses 30 gallons of water. How much water is used in a shower lasting 11 minutes?. The distance required to stop a car varies directly as the square of its speed. If 00 feet are required to stop a car traveling 60 miles per hour, how many feet are required to stop a car traveling 100 miles per hour? Inverse Variation If a situation is described by an equation in the form k y x where k is a nonzero constant, we say that y varies inversely as x. Practice: 1. The length of a violin string varies inversely as the frequency of its vibrations. A violin string 8 inches long vibrates at a frequency of 640 cycles per second. What is the frequency of a 10-inch string? Math Analysis Notes prepared by Mrs. Atkinson 1

22 Combined Variation Some problems will use both direct variation and inverse variation to relate two variables. When this happens you will only have one k value. If a situation is described by an equation in the form kz y x where k is a nonzero constant, we say that y varies directly with z and inversely with x. Practice: 1. The number of minutes needed to solve an exercise set of variation problems varies directly as the number of problems and inversely as the number of people working to solve the problems. It takes 4 people 3 minutes to solve 16 problems. How many minutes will it take 8 people to solve 4 problems? Joint Variation If a situation is described by an equation in the form y kxwz where k is a nonzero constant, we say that y varies jointly with x, w, and z. Practice: 1. The volume of a cone, V, varies jointly as its height, h, and the square of its radius, r. A cone with a radius measuring 6 feet and a height measuring 10 feet has a volume of 10π cubic feet. Find the volume of a cone having a radius of 1 feet and a height of feet. Math Analysis Notes prepared by Mrs. Atkinson