Polynomial and Rational Functions

Transcription

1 Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan; credit memor cards : modification of work b Paul Hudson) Chapter Outline. Comple Numbers. Quadratic Functions. Power Functions and Polnomial Functions. Graphs of Polnomial Functions. Dividing Polnomials. Zeros of Polnomial Functions.7 Rational Functions.8 Inverses and Radical Functions.9 Modeling Using Variation Introduction Digital photograph has dramaticall changed the nature of photograph. No longer is an image etched in the emulsion on a roll of film. Instead, nearl ever aspect of recording and manipulating images is now governed b mathematics. An image becomes a series of numbers, representing the characteristics of light striking an image sensor. When we open an image file, software on a camera or computer interprets the numbers and converts them to a visual image. Photo editing software uses comple polnomials to transform images, allowing us to manipulate the image in order to crop details, change the color palette, and add special effects. Inverse functions make it possible to convert from one file format to another. In this chapter, we will learn about these concepts and discover how mathematics can be used in such applications. This OpenSta book is available for free at 97

2 CHAPTER Polnomial and Rational Functions Learning Objectives In this section, ou will: Identif power functions. Identif end behavior of power functions. Identif polnomial functions. Identif the degree and leading coefficient of polnomial functions.. Power Functions and Polnomial Functions Figure (credit: Jason Ba, Flickr) Suppose a certain species of bird thrives on a small island. Its population over the last few ears is shown in Table. Year Bird Population ,08,9 Table The population can be estimated using the function P(t) = 0.t + 97t + 800, where P(t) represents the bird population on the island t ears after 009. We can use this model to estimate the maimum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will eamine functions that we can use to estimate and predict these tpes of changes. Identifing Power Functions In order to better understand the bird problem, we need to understand a specific tpe of function. A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fied real number. (A number that multiplies a variable raised to an eponent is known as a coefficient.) As an eample, consider functions for area or volume. The function for the area of a circle with radius r is A(r) = πr and the function for the volume of a sphere with radius r is V(r) = πr Both of these are eamples of power functions because the consist of a coefficient, π or _ π, multiplied b a variable r raised to a power. power function A power function is a function that can be represented in the form f () = k p where k and p are real numbers, and k is known as the coefficient.

3 SECTION. Power Functions and Polnomial Functions Q & A Is f () = a power function? No. A power function contains a variable base raised to a fied power. This function has a constant base raised to a variable power. This is called an eponential function, not a power function. Eample Identifing Power Functions Which of the following functions are power functions? f () = f () = f () = f () = Constant function Identif function Quadratic function Cubic function f () = Reciprocal function f () = f () = f () = Solution All of the listed functions are power functions. Reciprocal squared function Square root function Cube root function The constant and identit functions are power functions because the can be written as f () = 0 and f () = respectivel. The quadratic and cubic functions are power functions with whole number powers f () = and f () =. The reciprocal and reciprocal squared functions are power functions with negative whole number powers because the can be written as f () = and f () =. The square and cube root functions are power functions with fractional powers because the can be written as f () = / or f () = /. Tr It # Which functions are power functions? f () = g() = + h() = _ + Identifing End Behavior of Power Functions Figure shows the graphs of f () =, g() = and h() =, which are all power functions with even, whole-number powers. Notice that these graphs have similar shapes, ver much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper awa from the origin. f () = g() = h() = Figure Even-power functions This OpenSta book is available for free at

4 CHAPTER Polnomial and Rational Functions To describe the behavior as numbers become larger and larger, we use the idea of infinit. We use the smbol for positive infinit and for negative infinit. When we sa that approaches infinit, which can be smbolicall written as, we are describing a behavior; we are saing that is increasing without bound. With the even-power function, as the input increases or decreases without bound, the output values become ver large, positive numbers. Equivalentl, we could describe this behavior b saing that as approaches positive or negative infinit, the f () values increase without bound. In smbolic form, we could write as ±, f () Figure shows the graphs of f () =, g() =, and h() = 7, which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper awa from the origin. g() = f () = h() = 7 Figure Odd-power functions These eamples illustrate that functions of the form f () = n reveal smmetr of one kind or another. First, in Figure we see that even functions of the form f () = n, n even, are smmetric about the -ais. In Figure we see that odd functions of the form f () = n, n odd, are smmetric about the origin. For these odd power functions, as approaches negative infinit, f () decreases without bound. As approaches positive infinit, f () increases without bound. In smbolic form we write as, f () as, f () The behavior of the graph of a function as the input values get ver small ( ) and get ver large ( ) is referred to as the end behavior of the function. We can use words or smbols to describe end behavior. Figure shows the end behavior of power functions in the form f () = k n where n is a non-negative integer depending on the power and the constant. Even power Odd power Positive constant k > 0, f () and, f (), f () and, f () Negative constant k < 0, f () and, f () Figure, f () and, f ()

5 SECTION. Power Functions and Polnomial Functions 7 How To Given a power function f () = k n where n is a non-negative integer, identif the end behavior.. Determine whether the power is even or odd.. Determine whether the constant is positive or negative.. Use Figure to identif the end behavior. Eample Identifing the End Behavior of a Power Function Describe the end behavior of the graph of f () = 8. Solution The coefficient is (positive) and the eponent of the power function is 8 (an even number). As approaches infinit, the output (value of f ()) increases without bound. We write as, f (). As approaches negative infinit, the output increases without bound. In smbolic form, as, f (). We can graphicall represent the function as shown in Figure. Eample Figure Identifing the End Behavior of a Power Function Describe the end behavior of the graph of f () = 9. Solution The eponent of the power function is 9 (an odd number). Because the coefficient is (negative), the graph is the reflection about the -ais of the graph of f () = 9. Figure shows that as approaches infinit, the output decreases without bound. As approaches negative infinit, the output increases without bound. In smbolic form, we would write as, as, f () f () Figure This OpenSta book is available for free at

6 8 CHAPTER Polnomial and Rational Functions Analsis We can check our work b using the table feature on a graphing utilit. f () 0,000,000,000,9, 0 0,9, 0,000,000,000 Table We can see from Table that, when we substitute ver small values for, the output is ver large, and when we substitute ver large values for, the output is ver small (meaning that it is a ver large negative value). Tr It # Describe in words and smbols the end behavior of f () =. Identifing Polnomial Functions An oil pipeline bursts in the Gulf of Meico, causing an oil slick in a roughl circular shape. The slick is currentl miles in radius, but that radius is increasing b 8 miles each week. We want to write a formula for the area covered b the oil slick b combining two functions. The radius r of the spill depends on the number of weeks w that have passed. This relationship is linear. r(w) = + 8w We can combine this with the formula for the area A of a circle. A(r) = πr Composing these functions gives a formula for the area in terms of weeks. Multipling gives the formula. A(w) = A(r(w)) = A( + 8w) = π( + 8w) A(w) = 7π + 8πw + πw This formula is an eample of a polnomial function. A polnomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power. polnomial functions Let n be a non-negative integer. A polnomial function is a function that can be written in the form f () = a n n a + a + a 0 This is called the general form of a polnomial function. Each a i is a coefficient and can be an real number, but a n cannot = 0. Each product a i i is a term of a polnomial function. Eample Identifing Polnomial Functions Which of the following are polnomial functions? f () = + g() = ( ) h() = + Solution The first two functions are eamples of polnomial functions because the can be written in the form f () = a n n a + a + a 0, where the powers are non-negative integers and the coefficients are real numbers.

7 SECTION. Power Functions and Polnomial Functions 9 f () can be written as f () = +. g() can be written as g() = +. h() cannot be written in this form and is therefore not a polnomial function. Identifing the Degree and Leading Coefficient of a Polnomial Function Because of the form of a polnomial function, we can see an infinite variet in the number of terms and the power of the variable. Although the order of the terms in the polnomial function is not important for performing operations, we tpicall arrange the terms in descending order of power, or in general form. The degree of the polnomial is the highest power of the variable that occurs in the polnomial; it is the power of the first variable if the function is in general form. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term. terminolog of polnomial functions We often rearrange polnomials so that the powers are descending. Leading coefficient Degree f () = a n n + + a + a + a 0 Leading term When a polnomial is written in this wa, we sa that it is in general form. How To Given a polnomial function, identif the degree and leading coefficient.. Find the highest power of to determine the degree function.. Identif the term containing the highest power of to find the leading term.. Identif the coefficient of the leading term. Eample Identifing the Degree and Leading Coefficient of a Polnomial Function Identif the degree, leading term, and leading coefficient of the following polnomial functions. f () = + g(t) = t t + 7t h(p) = p p Solution For the function f (), the highest power of is, so the degree is. The leading term is the term containing that degree,. The leading coefficient is the coefficient of that term,. For the function g(t), the highest power of t is, so the degree is. The leading term is the term containing that degree, t. The leading coefficient is the coefficient of that term,. For the function h(p), the highest power of p is, so the degree is. The leading term is the term containing that degree, p ; the leading coefficient is the coefficient of that term,. Tr It # Identif the degree, leading term, and leading coefficient of the polnomial f () = +. This OpenSta book is available for free at

8 0 CHAPTER Polnomial and Rational Functions Identifing End Behavior of Polnomial Functions Knowing the degree of a polnomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polnomial function. Because the power of the leading term is the highest, that term will grow significantl faster than the other terms as gets ver large or ver small, so its behavior will dominate the graph. For an polnomial, the end behavior of the polnomial will match the end behavior of the term of highest degree. See Table. Polnomial Function Leading Term Graph of Polnomial Function f () = + f () = + + f () = + + f () = Table

9 SECTION. Power Functions and Polnomial Functions Eample Identifing End Behavior and Degree of a Polnomial Function Describe the end behavior and determine a possible degree of the polnomial function in Figure 7. Figure 7 Solution As the input values get ver large, the output values f () increase without bound. As the input values get ver small, the output values f () decrease without bound. We can describe the end behavior smbolicall b writing as, as, f () f () In words, we could sa that as values approach infinit, the function values approach infinit, and as values approach negative infinit, the function values approach negative infinit. We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polnomial creating this graph must be odd and the leading coefficient must be positive. Tr It # Describe the end behavior, and determine a possible degree of the polnomial function in Figure 8. Figure 8 Eample 7 Identifing End Behavior and Degree of a Polnomial Function Given the function f () = ( )( + ), epress the function as a polnomial in general form, and determine the leading term, degree, and end behavior of the function. Solution Obtain the general form b epanding the given epression for f (). f () = ( )( + ) = ( + ) = 9 + The general form is f () = 9 +. The leading term is ; therefore, the degree of the polnomial is. The degree is even () and the leading coefficient is negative ( ), so the end behavior is as, as, f () f () This OpenSta book is available for free at

10 CHAPTER Polnomial and Rational Functions Tr It # Given the function f () = 0.( )( + )( ), epress the function as a polnomial in general form and determine the leading term, degree, and end behavior of the function. Identifing Local Behavior of Polnomial Functions In addition to the end behavior of polnomial functions, we are also interested in what happens in the middle of the function. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing. We are also interested in the intercepts. As with all functions, the -intercept is the point at which the graph intersects the vertical ais. The point corresponds to the coordinate pair in which the input value is zero. Because a polnomial is a function, onl one output value corresponds to each input value so there can be onl one -intercept (0, a 0 ). The -intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one -intercept. See Figure 9. Turning point -intercepts Turning point -intercept Figure 9 intercepts and turning points of polnomial functions A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The -intercept is the point at which the function has an input value of zero. The -intercepts are the points at which the output value is zero. How To Given a polnomial function, determine the intercepts.. Determine the -intercept b setting = 0 and finding the corresponding output value.. Determine the -intercepts b solving for the input values that ield an output value of zero. Eample 8 Determining the Intercepts of a Polnomial Function Given the polnomial function f () = ( )( + )( ), written in factored form for our convenience, determine the - and -intercepts. Solution The -intercept occurs when the input is zero so substitute 0 for. The -intercept is (0, 8). f (0) = (0 )(0 + )(0 ) = ( )()( ) = 8

11 SECTION. Power Functions and Polnomial Functions The -intercepts occur when the output is zero. 0 = ( )( + )( ) = 0 or + = 0 or = 0 = or = or = The -intercepts are (, 0), (, 0), and (, 0). We can see these intercepts on the graph of the function shown in Figure 0. -intercept (0, 8) intercepts (, 0), (, 0), and (, 0) Eample 9 Figure 0 Determining the Intercepts of a Polnomial Function with Factoring Given the polnomial function f () =, determine the - and -intercepts. Solution The -intercept occurs when the input is zero. The -intercept is (0, ). f (0) = (0) (0) = The -intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polnomial. f () = = ( 9)( + ) = ( )( + )( + ) 0 = ( )( + )( + ) = 0 or + = 0 or + = 0 = or = or (no real solution) The -intercepts are (, 0) and (, 0). We can see these intercepts on the graph of the function shown in Figure. We can see that the function is even because f () = f ( ) Figure -intercepts (, 0) and (, 0) -intercept (0, ) Tr It # Given the polnomial function f () = 0, determine the - and -intercepts. This OpenSta book is available for free at

12 CHAPTER Polnomial and Rational Functions Comparing Smooth and Continuous Graphs The degree of a polnomial function helps us to determine the number of -intercepts and the number of turning points. A polnomial function of n th degree is the product of n factors, so it will have at most n roots or zeros, or -intercepts. The graph of the polnomial function of degree n must have at most n turning points. This means the graph has at most one fewer turning point than the degree of the polnomial or one fewer than the number of factors. A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must alwas occur at rounded curves. The graphs of polnomial functions are both continuous and smooth. intercepts and turning points of polnomials A polnomial of degree n will have, at most, n -intercepts and n turning points. Eample 0 Determining the Number of Intercepts and Turning Points of a Polnomial Without graphing the function, determine the local behavior of the function b finding the maimum number of -intercepts and turning points for f () = Solution Tr It #7 The polnomial has a degree of 0, so there are at most 0 -intercepts and at most 0 = 9 turning points. Without graphing the function, determine the maimum number of -intercepts and turning points for f () = Eample Drawing Conclusions about a Polnomial Function from the Graph What can we conclude about the polnomial represented b the graph shown in Figure based on its intercepts and turning points? Figure Solution The end behavior of the graph tells us this is the graph of an even-degree polnomial. See Figure. -intercepts Turning points Figure The graph has -intercepts, suggesting a degree of or greater, and turning points, suggesting a degree of or greater. Based on this, it would be reasonable to conclude that the degree is even and at least.

13 SECTION. Power Functions and Polnomial Functions Tr It #8 What can we conclude about the polnomial represented b the graph shown in Figure based on its intercepts and turning points? Figure Eample Drawing Conclusions about a Polnomial Function from the Factors Given the function f () = ( + )( ), determine the local behavior. Solution The -intercept is found b evaluating f (0). f (0) = (0)(0 + )(0 ) = 0 The -intercept is (0, 0). The -intercepts are found b determining the zeros of the function. 0 = ( + )( ) = 0 or + = 0 or = 0 = 0 or = or = The -intercepts are (0, 0), (, 0), and (, 0). The degree is so the graph has at most turning points. Tr It #9 Given the function f () = 0.( )( + )( ), determine the local behavior. Access these online resources for additional instruction and practice with power and polnomial functions. Find Ke Information about a Given Polnomial Function ( End Behavior of a Polnomial Function ( Turning Points and -Intercepts of Polnomial Functions ( Least Possible Degree of a Polnomial Function ( This OpenSta book is available for free at

14 CHAPTER Polnomial and Rational Functions. section EXERCISES Verbal. Eplain the difference between the coefficient of a power function and its degree.. In general, eplain the end behavior of a power function with odd degree if the leading coefficient is positive.. What can we conclude if, in general, the graph of a polnomial function ehibits the following end behavior? As, f () and as, f ().. If a polnomial function is in factored form, what would be a good first step in order to determine the degree of the function?. What is the relationship between the degree of a polnomial function and the maimum number of turning points in its graph? Algebraic For the following eercises, identif the function as a power function, a polnomial function, or neither.. f () = 7. f () = ( ) 8. f () = 9. f () = 0. f () = ( + )( ). f () = + For the following eercises, find the degree and leading coefficient for the given polnomial ( )( + ). ( ) For the following eercises, determine the end behavior of the functions. 7. f () = 8. f () = 9. f () = 0. f () = 9. f () = +. f () = +. f () = ( + ). f () = ( ) 7 For the following eercises, find the intercepts of the functions.. f (t) = (t )(t + )(t ). g(n) = (n )(n + ) 7. f () = 8. f () = f () = ( 8) 0. f () = ( + )( ) Graphical For the following eercises, determine the least possible degree of the polnomial function shown....

15 SECTION. Section Eercises For the following eercises, determine whether the graph of the function provided is a graph of a polnomial function. If so, determine the number of turning points and the least possible degree for the function This OpenSta book is available for free at

16 8 CHAPTER Polnomial and Rational Functions Numeric For the following eercises, make a table to confirm the end behavior of the function.. f () = 7. f () = 8. f () = ( ) 9. f () = ( )( )( ) 0. f () = 0 Technolog For the following eercises, graph the polnomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.. f () = ( ). f () = ( )( + ). f () = ( )(0 ). f () = ( )(0 ). f () =. f () = 7 7. f () = 8 8. f () = f () = 0. f () = 0.0 Etensions For the following eercises, use the information about the graph of a polnomial function to determine the function. Assume the leading coefficient is or. There ma be more than one correct answer.. The -intercept is (0, ). The -intercepts are (, 0), (, 0). Degree is. End behavior: as, f (), as, f ().. The -intercept is (0, 9). The -intercepts are (, 0), (, 0). Degree is. End behavior: as, f (), as, f ().. The -intercept is (0, 0). The -intercepts are (0, 0), (, 0). Degree is. End behavior: as, f (), as, f ().. The -intercept is (0, ). The -intercept is (, 0). Degree is. End behavior: as, f (), as, f ().. The -intercept is (0, ). There is no -intercept. Degree is. End behavior: as, f (), as, f (). Real-World ApplICATIOns For the following eercises, use the written statements to construct a polnomial function that represents the required information.. An oil slick is epanding as a circle. The radius of the circle is increasing at the rate of 0 meters per da. Epress the area of the circle as a function of d, the number of das elapsed. 7. A cube has an edge of feet. The edge is increasing at the rate of feet per minute. Epress the volume of the cube as a function of m, the number of minutes elapsed. 8. A rectangle has a length of 0 inches and a width of inches. If the length is increased b inches and the width increased b twice that amount, epress the area of the rectangle as a function of. 70. A rectangle is twice as long as it is wide. Squares of side feet are cut out from each corner. Then the sides are folded up to make an open bo. Epress the volume of the bo as a function of the width (). 9. An open bo is to be constructed b cutting out square corners of -inch sides from a piece of cardboard 8 inches b 8 inches and then folding up the sides. Epress the volume of the bo as a function of.