3.1 Graphing Quadratic Functions. Quadratic functions are of the form.

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1 3.1 Graphing Quadratic Functions A. Quadratic Functions Completing the Square Quadratic functions are of the form. 3. It is easiest to graph quadratic functions when the are in the form using transformations. Here, the parabola has the verte at. *Graph each of the following parabolas: A note about Aleks: Sometimes Aleks will ask ou to plot 5 points when graphing a parabola. Obviousl, one will be the verte. Usuall ou can use the leading coefficient of the parabola to help ou determine a second and third point easil. Remember to use the smmetr of the parabola to help find addtional points. Eample: 41

2 *For each of the following functions, complete the square to write each in the form. State the verte. Also find the -intercepts of each (ou ma wish to factor or use the quadratic formula)

3 B. Quadratic Functions The Verte Formula The quadratic function has its verte at. *Use the verte formula to find the verte of each of the following functions: 9. A note about vocabular of quadratic functions: If is an -intercept of the graph of it can also be said that is a solution or of the equation Further, and is a of can be called a of 3.1 #47 A projectile is thrown upward with an initial velocit of 176 ft/sec. After t seconds, its height above the ground is given b the function. a. Find the projectile's height above the ground after 2 seconds. 10. b. Sketch the graph modeling the projectile's height. c. What is the projectile's maimum height? What is the value of t at this height? 11. d. How man seconds after it is thrown will the projectile strike the ground? 43

4 ALEKS Problem Find a quadratic function f whose graph is shown below. 44

5 3.2 Long & Snthetic Division 3. Divide: Long division alwas works; snthetic division onl works when dividing b factors (those without eponents). 1. Divide: Snthetic division can also be used to evaluate polnomials: 4. If, find in two was. 2. Divide using snthetic division: 45

6 5. Use snthetic division to determine whether is a factor of 6. Write a polnomial of least degree having the given zeroes (roots, solutions): a. b. 3.2 #66 Given that is a zero of, write in a completel factored form. c. d. 3.2 #70 Given that is a zero of completel factored form., write in a 3.2 #72 Factor: 46

7 3.2 #77 Factor: 3.3 Factoring Higher-Degreed Polnomials The Rational Zero Theorem: If has rational zeroes, then the are of the form where a is the set of factors of the leading coefficient, and b is the set of factors of the constant term. *List all possible rational zeroes of Theorem: If the sum of the coefficients is, then is a zero (and is a factor). If after changing the signs of the coefficients of the odd-degreed terms, the sum of the "new" coefficients is zero, then is a zero. Returning to our previous problem, factor: Factor: We need some new tools to break this down further... 47

8 The Intermediate Value Theorem: If is a polnomial and, and and, then there eists some number "c" in such that 3.3 #44 Factor. Find the zeroes. picture: 3.3 #30 Use the Intermediate Value Theorem to determine whether a zero eists for on the interval. 3.3 #49 Factor. Find the zeroes. 48

9 Descartes' Rule of Signs: Given the polnomial, the number of positive real zeroes is equal to the number of variations in sign of, or less that that b subtracting twos. The number of negative real zeroes is equal to the number of variations in sign of, or less that that b subtracting twos. 2. Theorem: If is a polnomial with real coefficients, then if there are an comple zeroes, the will be comple conjugate pairs. Possible positive zeroes Possible negative zeroes Possible comple zeroes Total number of zeroes *Use Descartes' Rule of Signs to count the number of possible positive, negative and comple zeroes (roots) of each: 1. Upper and Lower Bounds Propert: Given is a polnomial with real coefficients. Possible positive zeroes Possible negative zeroes Possible comple zeroes Total number of zeroes 1. If is divided b using snthetic division and all coefficients in the quotient row are either positive or zero, then is an upper bound on the zeroes of P. 2. If is divided b using snthetic division and all coefficients in the quotient row alternate in sign, then is a lower bound on the zeroes of P. *For both 1 and 2, zero coefficients can be either positive or negative as needed. 49

10 3.3 #89 Use the Rational Zero Theorem and Descartes' Rule of Signs along with the tests for 1 and to find all the zeroes of 3.3 #96 Use the Rational Zero Theorem and Descartes' Rule of Signs along with the tests for 1 and to find all the zeroes of 50

11 ALEKS Problem (requires the use of the Aleks graphing calculator) A 2 foot thick slice is cut off the top of a cube, resulting in a rectangular bo that has volume. Use the ALEKS graphing calculator to find the side length of the original cube. Round our answer to two decimal places. ALEKS Problem Suppose that the polnomial has real coefficients with Suppose also that has the following zeroes: 2, 3, Using this information, answer the following questions: a. What is another zero of? b. At most, how man real zeroes of are there? c. At most, how man imaginar zeroes of are there? ALEKS Problem (requires the use of the Aleks graphing calculator) The width of a rectangular bo is 2 times its length, and its height is 4 ft more than its length. The volume of the bo is. Use the ALEKS graphing calculator to find the length of the bo. Round our answer to two decimal places. ALEKS Problem Find all other zeroes of the polnomial, given that is a zero. 51

12 3.4 Graphing Polnomial Functions Theorem: A polnomial of degree has at most vertices (or less than that b subtracting twos). There are two tpes of -intercepts: Cut and Bounce: A "bounce" -intercept occurs when the zero has an multiplicit; a "cut" occurs when the zero has an multiplicit. 52

13 Sketch: ALEKS Problem Below is the graph of a polnomial function with real coefficients. Use the graph to answer the questons about. All local etrema of are shown in the graph. Sketch: Hint: The graph contains the following points: a. The function is increasing over which intervals? b. The function has minima at which - values? c. What is the sign of the leading coefficient of? d. Which of the following is a possibilit for the degree of? Check (circle) all that appl

14 54

15 3.5, 3.6 Graphing Rational Functions Rational functions are of the form where and have no Graph and then write a piecewise-defined fucntion which would make into a continuous function. factors in common. If and DO have factors in common, ou get removable discontinuities in our graph. For eample: Three Tpes of Discontinuities: removable discontinuities We can "fill the gap" b creating a piecewisedefined function: nonremovable (gap) discontinuities nonremovable (asmptotic) discontinuities 55

16 Steps for Graphing Rational Functions: Graph: 1. Set the numerator equal to zero to find the -intercept. 2. Set the denominator equal to zero to determine locations of an vertical asmptotes. 3. Look at the degrees of the numerator and denominator... a. If the degree of the denominator is higher, then there is a horizontal asmptote at, the -ais. b. If the degree of the numerator is higher, then do long division to determine the diagonal (oblique) or parabolic asmptote. c. If the degree of the numerator and the denominator is the same, then there is a horizontal asmptote at the ratio of the leading coefficients. Graph: 4. Use an - chart to plot additonal points. Etra note: If the denominator contains an even eponent, then the vertical asmptote which it corresponds to has curves on both sides that approach the same infinit. If each vertical asmptote referred to b the denominator has an odd multiplicit, then the curve approaches opposite infinities on either side of the vertical asmptote. 56

17 Graph: Graph: 57

18 Graph: Graph: 58

19 3.7 Polnomial and Rational Inequalities 3.7 #42 Solve: Recall... When is? 3.7 #10 where is 3.7 #54 Solve: 3.7 #14 Solve: 3.7 #20 Solve: 59

20 Find the domain of Solve: Solve: 60

21 3.8 Variation Direct Variation Solve the following: 6. m varies directl as the square of. If m=200 when =20, find m when =32. Inverse Variation "k" is called the * Write a variation model for each: Aleks Problems: 1. W varies directl as Z. 2. m varies inversel as t. 3. w varies jointl as p and f. Find the value of k for each m varies directl as and when is 8, m is T is inversel proportional to and when is 40, T is

22 Chapter 3 Review All Aleks Problems 1. 62

23 2. For the polnomial below, 3 is a zero. Epress g() as a product of linear factors. g 3 2 ( ) Use the rational zero theorem to list all possible rational zeros of the following. h( ) Find the equation of the quadratic function f whose graph is shown below. 5. Write an equation that epresses the following relationship. Use k as the constant of proportionalit. P varies jointl with the square of d and the cube of u. 63

24 6. The function below has at least one rational zero. Use this fact to find all zeros of the function. 7. Graph the rational function f( ) f 3 2 ( ) On a given planet, the weight of an object varies directl with the mass of the object. Suppose that an object whose mass is 9 kg weighs 90 N. Calculate the mass of another object that weighs 40 N. 64

25 9. Graph the parabola Plot the verte and four additional points, two on each side of the verte Graph the parabola Plot the verte and four additional points, two on each side of the verte. 65

26 11. For the polnomial below, 1 is a zero. Epress h() as a product of linear factors. h 3 2 ( ) Sketch 2 1 f( ) Sketch: Use Descartes' Rule of Signs to determine the number of possible positive real zeros and negative real zeros of f ( ) Solve

27 4.1 Functions and Thier Inverses A function is if it passes both a vertical and horizontal line test. If a function is one-to-one, then it is (it has an inverse which is also a function). *Find the inverse of each function and then graph each on the same graph: 1. To find the inverse of a function from its equation, switch the and, and then solve for the "new". To find the inverse of a function from its graph, reflect the graph across the line. If is on, then is on the graph of its inverse. *Given the graph of, graph its inverse

28 3. Aleks Problem: The one-to-one functions and are defined as follows: Find the following: To prove algebraicall that two functions are inverses of each other, show that and. Aleks Problem: The one-to-one functions defined b. Find, the inverse of. Then, give the domain and range of using interval notation. is 4.1 #57 Prove that and are inverses of each other. 68

29 4.2, 4.3 Eponential & Logarithmic Functions Graph is an eponential equation. Graph: Calculators: Eponents and Logarithms Eponential kes: Convert each from logarithmic form to eponential form (or vice versa): Logarithmic kes: Logarithmic Form Eponential Form Definition: As On our calculator, find the following values:

30 5. Aleks Problem: Graph the function: 6. Change-of-Base Formula: Solve each equation: Evaluate each of the following:

31 4.4 Solving Ep. & Log. Equations 4.2 #57 Solve: Properties of Logarithms: #67 Solve: 7. *Write each as separate, simplified logarithms: #57 Solve: 71

32 Solve: 4.4 #73 Solve: Solve: 4.4 #73 Solve: 72

33 4.4 #83 Solve: Aleks Problem: Consider the equation. Find the value of. Round our answer to 3 decimal places 4.4 #22 Solve: 73

34 4.5 Applications of Ep. & Log Equations Compound interest: For how long should $1000 be invested at 1.1% compounded dail for the mone to double? $1000 is invested at 1.2% for 10 ears. Find the value of the account after 10 ears if the interest is compounded... a. quarterl. b. monthl. c. dail. d. continuousl. 74

35 If $1500 is invested at 2.25%, compounded monthl, how long would it take to grow into $1800? How much should be invested at 1.1%, compounded dail, to grow to $2000 in 6 months? Annuit Formula (optional, if time) If $100 is invested ever month into an account which earns 8.5% interest, compounded monthl, for 45 ears, what would the balance of the account be at the end of the 45 ears? For how long should $10,000 be invested at 2.5% compounded dail in order for the mone to triple? nt Pn r Use A 1 1, where P is the r n amount deposited n times per ear. 75

36 Chapters 3 & 4 Mied Review 1. Solve: 3. Graph: 2. Given that is a zero, find the other zeroes of. 76

37 4. Graph: 6. Graph 7. Find all the zeroes of 5. Graph: 77

38 8. Sketch: 11. Solve: 9. Solve: 12. Solve: 10. Solve: 78

39 13. Solve: 15. For continuousl compounding interest, at what interest rate should $500 be invested so that is grows to $750 in 8 ears? 16. Find the verte: 14. Solve: 17. Graph 79

40 18. For how long should $800 be invested at 4.3%, compounded dail, in order for it to grow to $2000? 20. Use the Rational Zero Theorem and Descartes' Rule of Signs along with the tests for 1 and to find all the zeroes of 19. Find and graph the inverse of 80

41 21. Graph f( ) For f( ), find f 8 5 domain and range of f 1 ( ). 1 ( ), and the 22. Write a piece-wise defined function to make f() (from the previous problem) continuous. 81

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