Essential Question How many turning points can the graph of a polynomial function have?

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1 .8 Analzing Graphs of Polnomial Functions Essential Question How man turning points can the graph of a polnomial function have? A turning point of the graph of a polnomial function is a point on the graph at which the function changes from increasing to decreasing, or decreasing to increasing. turning point.5 turning point Approimating Turning Points ATTENDING TO PRECISION To be proficient in math, ou need to epress numerical answers with a degree of precision appropriate for the problem contet. Work with a partner. Match each polnomial function with its graph. Eplain our reasoning. Then use a graphing calculator to approimate the coordinates of the turning points of the graph of the function. Round our answers to the nearest hundredth. a. f() = + b. f() = + + c. f() = + d. f() = + 5 e. f() = + f. f() = A. B. C. D. E. F. 7 Communicate Your Answer. How man turning points can the graph of a polnomial function have?. Is it possible to sketch the graph of a cubic polnomial function that has no turning points? Justif our answer. Section.8 Analzing Graphs of Polnomial Functions

2 .8 Lesson What You Will Learn Core Vocabular local maimum, p. local minimum, p. even function, p. 5 odd function, p. 5 Previous end behavior increasing decreasing smmetric about the -ais Use -intercepts to graph polnomial functions. Use the Location Principle to identif zeros of polnomial functions. Find turning points and identif local maimums and local minimums of graphs of polnomial functions. Identif even and odd functions. Graphing Polnomial Functions In this chapter, ou have learned that zeros, factors, solutions, and -intercepts are closel related concepts. Here is a summar of these relationships. Concept Summar Zeros, Factors, Solutions, and Intercepts Let f() = a n n + a n n + + a + a 0 be a polnomial function. The following statements are equivalent. Zero: k is a zero of the polnomial function f. Factor: k is a factor of the polnomial f(). Solution: k is a solution (or root) of the polnomial equation f() = 0. -Intercept: If k is a real number, then k is an -intercept of the graph of the polnomial function f. The graph of f passes through (k, 0). Using -Intercepts to Graph a Polnomial Function Graph the function f() = ( + )( ). SOLUTION Step Plot the -intercepts. Because and are zeros of f, plot (, 0) and (, 0). (, 0) Step Plot points between and beond the -intercepts. 0 8 (, 0) Step Determine end behavior. Because f() has three factors of the form k and a constant factor of, f is a cubic function with a positive leading coefficient. So, f() as and f() + as +. Step Draw the graph so that it passes through the plotted points and has the appropriate end behavior. Monitoring Progress Graph the function. Help in English and Spanish at BigIdeasMath.com. f() = ( + )( ). f() = ( + )( )( ) Chapter Polnomial Functions

3 The Location Principle You can use the Location Principle to help ou find real zeros of polnomial functions. Core Concept The Location Principle If f is a polnomial function, and a and b are two real numbers such that f(a) < 0 and f(b) > 0, then f has at least one real zero between a and b. To use this principle to locate real zeros of a polnomial function, find a value a at which the polnomial function is negative and another value b at which the function is positive. You can conclude that the function has at least one real zero between a and b. f(b) f(a) a b zero Check 5 0 Zero X=.5 Y=0 0 5 Find all real zeros of f() = SOLUTION Locating Real Zeros of a Polnomial Function Step Use a graphing calculator to make a table. Step Use the Location Principle. From the table shown, ou can see that f() < 0 and f() > 0. So, b the Location Principle, f has a zero between and. Because f is a polnomial function of degree, it has three zeros. The onl possible rational zero between and is. Using snthetic division, ou can confirm that is a zero. Step Write f() in factored form. Dividing f() b its known factor gives a quotient of + +. So, ou can factor f() as f() = ( ) ( + + ) = ( ) ( ) = ( ) ( + )( + ). From the factorization, there are three zeros. The zeros of f are,, and. Check this b graphing f. Monitoring Progress. Find all real zeros of f() = 8 +. X 0 5 X= Y Help in English and Spanish at BigIdeasMath.com Section.8 Analzing Graphs of Polnomial Functions

4 READING Local maimum and local minimum are sometimes referred to as relative maimum and relative minimum. Turning Points Another important characteristic of graphs of polnomial functions is that the have turning points corresponding to local maimum and minimum values. The -coordinate of a turning point is a local maimum of the function when the point is higher than all nearb points. The -coordinate of a turning point is a local minimum of the function when the point is lower than all nearb points. The turning points of a graph help determine the intervals for which a function is increasing or decreasing. function is decreasing local maimum function is increasing function is increasing local minimum Core Concept Turning Points of Polnomial Functions. The graph of ever polnomial function of degree n has at most n turning points.. If a polnomial function of degree n has n distinct real zeros, then its graph has eactl n turning points. Finding Turning Points Graph each function. Identif the -intercepts and the points where the local maimums and local minimums occur. Determine the intervals for which each function is increasing or decreasing. a. f() = + b. g() = SOLUTION 5 a. Use a graphing calculator to graph the function. The graph of f has one -intercept and two turning points. Use the graphing calculator s zero, maimum, and minimum features to approimate the coordinates of the points. 5 The -intercept of the graph is.0. The function has a local maimum at (0, ) and a local minimum at (, ). The function is increasing when < 0 and > and decreasing when 0 < <. Maimum X=0 Y= 0 0 b. Use a graphing calculator to graph the function. The graph of g has four -intercepts and three turning points. Use the graphing calculator s zero, maimum, and minimum features to approimate the coordinates of the points. Minimum X= Y= The -intercepts of the graph are., 0.9,.8, and 5.0. The function has a local maimum at (., 5.) and local minimums at ( 0.57,.5) and (.9,.0). The function is increasing when 0.57 < <. and >.9 and decreasing when < 0.57 and. < <.9. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Graph f() = Identif the -intercepts and the points where the local maimums and local minimums occur. Determine the intervals for which the function is increasing or decreasing. Chapter Polnomial Functions

5 Even and Odd Functions Core Concept Even and Odd Functions A function f is an even function when f( ) = f() for all in its domain. The graph of an even function is smmetric about the -ais. A function f is an odd function when f( ) = f() for all in its domain. The graph of an odd function is smmetric about the origin. One wa to recognize a graph that is smmetric about the origin is that it looks the same after a 80 rotation about the origin. Even Function Odd Function (, ) (, ) (, ) (, ) For an even function, if (, ) is on the For an odd function, if (, ) is on the graph, then (, ) is also on the graph. graph, then (, ) is also on the graph. Identifing Even and Odd Functions Determine whether each function is even, odd, or neither. a. f() = 7 b. g() = + c. h() = + SOLUTION a. Replace with in the equation for f, and then simplif. f( ) = ( ) 7( ) = + 7 = ( 7) = f() Because f( ) = f(), the function is odd. b. Replace with in the equation for g, and then simplif. g( ) = ( ) + ( ) = + = g() Because g( ) = g(), the function is even. c. Replacing with in the equation for h produces h( ) = ( ) + = +. Because h() = + and h() =, ou can conclude that h( ) h() and h( ) h(). So, the function is neither even nor odd. Monitoring Progress Determine whether the function is even, odd, or neither. Help in English and Spanish at BigIdeasMath.com 5. f() = + 5. f() = 5 7. f() = 5 Section.8 Analzing Graphs of Polnomial Functions 5

6 .8 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE A local maimum or local minimum of a polnomial function occurs at a point of the graph of the function.. WRITING Eplain what a local maimum of a function is and how it ma be different from the maimum value of the function. Monitoring Progress and Modeling with Mathematics ANALYZING RELATIONSHIPS In Eercises, match the function with its graph.. f() = ( )( )( + ). h() = ( + ) ( + ) 5. g() = ( + )( )( + ). f() = ( ) ( + ) A. B. ERROR ANALYSIS In Eercises 5 and, describe and correct the error in using factors to graph f. 5. f() = ( + )( ) C. D.. f() = ( ) In Eercises 7, graph the function. (See Eample.) 7. f() = ( ) ( + ) 8. f() = ( + ) ( + ) 9. h() = ( + ) ( )( ) 0. g() = ( + )( + )( ). h() = ( 5)( + )( ). g() = ( + )( + 8)( ). h() = ( )( + + ) In Eercises 7, find all real zeros of the function. (See Eample.) 7. f() = + 8. f() = + 9. h() = h() = 8. g() = f() = 5. f() = ( )( + ) Chapter Polnomial Functions

7 In Eercises 0, graph the function. Identif the -intercepts and the points where the local maimums and local minimums occur. Determine the intervals for which the function is increasing or decreasing. (See Eample.). g() = + 8. g() = + 5. h() = +. f() = f() = f() = h() = g() = In Eercises, estimate the coordinates of each turning point. State whether each corresponds to a local maimum or a local minimum. Then estimate the real zeros and find the least possible degree of the function The graph of f has -intercepts at =, =, and = 5. f has a local maimum value when =. f has a local minimum value when = and when =. In Eercises 9, determine whether the function is even, odd, or neither. (See Eample.) 9. h() = 7 0. g() = +. f() = +. f() = 5 +. g() = f() = f() =. h() = USING TOOLS When a swimmer does the breaststroke, the function S =t t 870t t 77t + t.t models the speed S (in meters per second) of the swimmer during one complete stroke, where t is the number of seconds since the start of the stroke and 0 t.. Use a graphing calculator to graph the function. At what time during the stroke is the swimmer traveling the fastest? 5.. OPEN-ENDED In Eercises 7 and 8, sketch a graph of a polnomial function f having the given characteristics. 7. The graph of f has -intercepts at =, = 0, and =. f has a local maimum value when =. f has a local minimum value when =. 8. USING TOOLS During a recent period of time, the number S (in thousands) of students enrolled in public schools in a certain countr can be modeled b S = ,00, where is time (in ears). Use a graphing calculator to graph the function for the interval 0. Then describe how the public school enrollment changes over this period of time. 9. WRITING Wh is the adjective local, used to describe the maimums and minimums of cubic functions, sometimes not required for quadratic functions? Section.8 Analzing Graphs of Polnomial Functions 7

8 50. HOW DO YOU SEE IT? The graph of a polnomial function is shown. = f() 0 0 a. Find the zeros, local maimum, and local minimum values of the function. b. Compare the -intercepts of the graphs of = f() and = f(). c. Compare the maimum and minimum values of the functions = f() and = f(). 5. PROBLEM SOLVING Quonset huts are temporar, all-purpose structures shaped like half-clinders. You have 00 square feet of material to build a quonset hut. a. The surface area S of a quonset hut is given b S = πr + πr. Substitute 00 for S and then write an epression for in terms of r. b. The volume V of a quonset hut is given b V = πr. Write an equation that gives V as a function in terms of r onl. c. Find the value of r that maimizes the volume of the hut. 5. MAKING AN ARGUMENT Your friend claims that the product of two odd functions is an odd function. Is our friend correct? Eplain our reasoning. 5. MODELING WITH MATHEMATICS You are making a rectangular bo out of a -inch-b-0-inch piece of cardboard. The bo will be formed b making the cuts shown in the diagram and folding up the sides. You want the bo to have the greatest volume possible. 0 in. in. a. How long should ou make the cuts? b. What is the maimum volume? c. What are the dimensions of the finished bo? 5. THOUGHT PROVOKING Write and graph a polnomial function that has one real zero in each of the intervals < <, 0 < <, and < < 5. Is there a maimum degree that such a polnomial function can have? Justif our answer. 55. MATHEMATICAL CONNECTIONS A clinder is inscribed in a sphere of radius 8 inches. Write an equation for the volume of the clinder as a function of h. Find the value of h that maimizes the volume of the inscribed clinder. What is the maimum volume of the clinder? h 8 in. Maintaining Mathematical Proficienc State whether the table displas linear data, quadratic data, or neither. Eplain. (Section.) 5. Months, 0 Savings (dollars), Reviewing what ou learned in previous grades and lessons Time (seconds), 0 Height (feet), Chapter Polnomial Functions