Graphing Polynomial Functions

Transcription

1 LESSON 7 Graphing Polnomial Functions Graphs of Cubic and Quartic Functions UNDERSTAND A parent function is the most basic function of a famil of functions. It preserves the shape of the entire famil. The first graph models the function f () 5 3, which is the most basic cubic function. The third graph models the function a() 5, which is the most basic of the quartic functions. Adding terms to the function and/or changing the leading coefficient can change the shape, orientation, and location of the graph of the function. f() 3 0 The degree and leading coefficient of a polnomial function can tell ou about the graph of a function. The influence the end behavior of the graph, which is the behavior of the graph as approaches positive infinit ( 1 `) or negative infinit ( `). a() 8 The end behavior of f (): As 1`, f () 1`, and as `, f () `. The end behavior of a(): As 1`, a() 1`, and as `, a() 1`. The degree can tell ou if the arms of the graph go in opposite directions or in the same direction. The degree of f () is odd, so the arms go in opposite directions. 0 g() 3 1 The degree of a() is even, so the arms of the graph go in the same direction. The leading coefficient can tell ou as approaches 1` whether the function will approach 1` or `. Graphicall, this means whether the right arm of the graph is pointing up or down. The leading coefficient of g() is positive so as 1`, g() 1`. This means the right arm of the graph is pointing up. The leading coefficient of c() is negative so as 1`, c() `. This means the right arm of the graph is pointing down. 0 c() 1 0 Duplicating this page is prohibited b law. 01 Triumph Learning, LLC Unit 1: Polnomial, Rational, and Radical Relationships

2 Connect Analze each function. Describe the end behavior of the function, determine the sign of the leading coefficient, whether it is even or odd degree, and the graph s parent function. a. b. 3 a() b() c. f () d. g() Describe the graph in part a. Duplicating this page is prohibited b law. 01 Triumph Learning, LLC 3 CHECK The end behavior is a() 1` as 1` and a() ` as `. Since the arms go in opposite directions, it is an odddegree function. The leading coefficient is positive because a() 1` as 1` and the right arm points up. Its parent function is a() 5 3, so it is a cubic function. Describe the function in part c. The end behavior can be described as follows: As approaches both 1` and `, f () 1`. Since f () 1` as 1` and the leading coefficient is positive, the right arm of the graph will point up. The leading term is raised to the th power, which means this is an even quartic function with f () 5 as the parent function. Wh is it true that f () 5 f () for all in the function from part c? Would it be possible for a cubic function to be even? Describe the graph in part b. The end behavior is b() ` as 1` and b() ` as `. Since the arms go in the same direction, it is an evendegree function. The leading coefficient is negative because b() ` as 1` and the right arm points down. Its parent function is b() 5, so it is a quartic function. Describe the function in part d. The end behavior can be described as follows: As 1`, g() `, and as `, g() 1`. Since g() ` as 1` and the leading coefficient is negative, the right arm of the graph will point down. The leading term is raised to the 3rd power, which means this is an odd cubic function with f () 5 3 as the parent function. Lesson 7: Graphing Polnomial Functions 3

3 Graphing and Factoring UNDERSTAND The roots of a polnomial are the zeros of its related function. These are the values for which f () 5 0. When the graph of a function equals 0, it touches the -ais. Therefore, the -intercepts of a graph give the zeros of the function. Not all zeros of polnomial functions are real numbers. The -intercepts show onl the real zeros of a function. Identifing the real roots of a function can help ou to factor it. Graphs of higher-order polnomials ma have several -intercepts. The number of times a graph crosses or touches the -ais tells ou the number of real roots it has. A quintic (5th-degree) and a quartic function are shown. f() g() The quintic function has arms that point in different directions, like cubic functions. This is true of all functions with an odd degree. This means that the range is (`, `). Since the range includes 0, the graph must cross the -ais at least once. We can conclude, then, that all functions with an odd degree have at least 1 real root. For functions with an even degree, such as the quartic function shown, both ends increase to ` or decrease to `. This means that the graph ma or ma not touch the -ais. The graph shown above, for eample, has no -intercept. We can conclude, then, that a function with an even degree ma or ma not have an real roots. It will, however, have at least one root that ma be comple. Then end behavior is the behavior of the graph as approaches positive infinit ( 1`) or negative infinit ( `). The leading coefficient of a polnomial function is a factor in determining the end behavior of a graph. The leading coefficient for f () is positive, so as 1`, f () 1`. For g(), the leading coefficient is negative, so as 1`, g () `. UNDERSTAND Factoring a polnomial function can help in sketching its graph b allowing ou to quickl find its -intercepts. Substitute 0 for to find its -intercept at (0, f (0)). You can also use other attributes to get a general idea about the shape of a graph: Look at the degree to determine the general shape. Look at the leading coefficient, along with the degree, to predict end behavior. Determine if a function is even or odd. This will tell ou if it is smmetrical about the -ais or the origin. Duplicating this page is prohibited b law. 01 Triumph Learning, LLC Unit 1: Polnomial, Rational, and Radical Relationships

4 Connect Find the zeros of the polnomial from its graph. Then, write the polnomial in factored form Identif the -intercepts. Find the binomial factors. The graph crosses the -ais at (5, 0), (, 0), and (, 0). The point (5, 0) means that 5 is a root. If a polnomial has root a, then ( a) is a factor of the polnomial. So, the root 5 corresponds to the factor [ (5)], or ( 1 5). 3 Write the polnomial in factored form. Since is a root, ( 1 ) is a factor of the polnomial. All three real zeros have been identified from the graph. Write the polnomial as the product of the factors associated with each zero. Since is a root, ( ) is a factor of the polnomial. Duplicating this page is prohibited b law. 01 Triumph Learning, LLC The graph shows zeros at 5,, and. The polnomial modeled b the graph is ( 1 )( )( 1 5). TRY Find the zeros of the polnomial from its graph Lesson 7: Graphing Polnomial Functions 5

5 EXAMPLE A Sketch the graph of the function f () Factor the polnomial to find the zeros of the function. Factor out 1 to get 1( ). Treat the epression as a quadratic in which the variable is. Let z 5, and rewrite the epression as a quadratic. 1(z 17z 1 1) 5 1(z 1)(z 1) Substitute back into the epression. 1( 1)( 1) The quadratic factors are both a difference of squares. The completel factored polnomial is 1( )( 1 )( 1 1)( 1). The zeros are, 1, 1, and. Find the intercepts of the graph. Each real zero of the function gives an -intercept of the graph. The -intercepts are (, 0), (1, 0), (1, 0), and (, 0). Find f (0) to find the -intercept. f (0) 5 (0) 1 17(0) The -intercept is (0, 1) 3 TRY Look at the leading term to determine the overall shape and end behavior of the graph. The leading term is. This means that both ends of the graph will point down. Sketch the graph of g() Sketch the graph Duplicating this page is prohibited b law. 01 Triumph Learning, LLC Unit 1: Polnomial, Rational, and Radical Relationships

6 EXAMPLE B Sketch the graph of the function f () Factor the polnomial to find the zeros of the function. Factor b grouping ( 3 1 ) 1 ( 1 ) 5 ( 1 1) 1 ( 1 1) 5 ( 1 )( 1 1) The quadratic factor has no real zeros. Comple zeros do not give -intercepts, so we do not need to find the comple zeros. The other factor, 1 1, indicates that 1 is a zero of the function. Find the intercepts. The function has a zero at 5 1, so it has an -intercept at (1, 0). Find f (0) to find the -intercept. f (0) 5 (0) 3 1 (0) 1 (0) 1 5 The -intercept is (0, ). 3 Look at the leading term to determine the overall shape and end behavior of the graph. Duplicating this page is prohibited b law. 01 Triumph Learning, LLC DISCUSS The leading term is 3. This means that the left arm will point down and the right arm will point up. Find the zeros of the polnomial using the rational roots theorem. What do the zeros tell ou about the graph? Sketch the graph Lesson 7: Graphing Polnomial Functions 7

7 Practice Describe the end behavior of each function As approaches `, f () approaches. As approaches 1`, f () approaches As approaches `, f () approaches. As approaches 1`, f () approaches As approaches `, f () approaches. As approaches 1`, f () approaches. Find the real zeros of each function Duplicating this page is prohibited b law. 01 Triumph Learning, LLC 8 Unit 1: Polnomial, Rational, and Radical Relationships

8 Choose the best answer.. Which of the following is not true of f () ? A. Its graph has at least one -intercept. B. It has at least one real root. C. As approaches 1`, f () approaches 1`. D. Its graph has a -intercept at (10, 0). 7. Which of the following is not true of g() ? A. Its graph has a -intercept at (0, 90). B. It has at least one real root. C. As approaches `, g () approaches `. D. As approaches `, g () approaches `. The graph of each function is shown. Write the function in factored form. Do not include comple numbers. 8. f () g() f () 5 g() 5 Duplicating this page is prohibited b law. 01 Triumph Learning, LLC Lesson 7: Graphing Polnomial Functions 9

9 Answer each question about the function. Then, sketch a graph of the function. 10. f () Write the function in factored form: f () 5 List the real zeros: List all -intercepts: List the -intercept: Describe the end behavior: As approaches `, f () approaches. As approaches `, f () approaches. Is the function even or odd? Sketch a graph of the function Solve. 11. MODEL Two different graphs have the same -intercepts. Find the equation of each function modeled in standard form. The graph of h () is shown. 0 h() The function is h() 5. The graph of j () is shown. 0 j() The function is j() 5. Duplicating this page is prohibited b law. 01 Triumph Learning, LLC 70 Unit 1: Polnomial, Rational, and Radical Relationships

10 1. COMPARE The graph of k() is shown Describe the graph, and compare it to a cubic function with the same leading coefficient. Duplicating this page is prohibited b law. 01 Triumph Learning, LLC Lesson 7: Graphing Polnomial Functions 71