2.3 Polynomial Functions of Higher Degree with Modeling

Transcription

1 SECTION 2.3 Polnomial Functions of Higher Degree with Modeling Polnomial Functions of Higher Degree with Modeling What ou ll learn about Graphs of Polnomial Functions End Behavior of Polnomial Functions Zeros of Polnomial Functions Intermediate Value Theorem Modeling... and wh These topics are important in modeling and can be used to provide approimations to more complicated functions, as ou will see if ou stud calculus. Graphs of Polnomial Functions As we saw in Section 2.1, a polnomial function of degree 0 is a constant function and graphs as a horizontal line. A polnomial function of degree 1 is a linear function; its graph is a slant line. A polnomial function of degree 2 is a quadratic function; its graph is a parabola. We now consider polnomial functions of higher degree. These include cubic functions (polnomials of degree 3) and quartic functions (polnomials of degree ). Recall that a polnomial function of degree n can be written in the form p12 = a n n + a n-1 n-1 + g + a a 1 + a 0, a n 0. Here are some important definitions associated with polnomial functions and this equation DEFINITION The Vocabular of Polnomials a n n, a n-1 n-1,..., a 0 is a term of the polnomial. is written in standard form. a n, a n-1,..., a 0 are the coefficients of the polnomial. a n n is the leading term, and a 0 is the constant term In Eample 1 we use the fact from Section 2.1 that the constant term a 0 of a polnomial function p is both the initial value of the function p102 and the -intercept of the graph to provide a quick and eas check of the transformed graphs FIGURE 2.19 The graphs of g12 = 1 + 1) 3 and ƒ12 = 3. The graphs of h12 = and ƒ12 =-. (Eample 1) 5 EXAMPLE 1 Graphing Transformations of Monomial Functions Describe how to transform the graph of an appropriate monomial function ƒ12 = a n n into the graph of the given function. Sketch the transformed graph b hand and support our answer with a grapher. Compute the location of the -intercept as a check on the transformed graph. g12 = h12 = You can obtain the graph of g12 = b shifting the graph of ƒ12 = 3 one unit to the left, as shown in Figure 2.19a. The -intercept of the graph of g is g102 = =, which appears to agree with the transformed graph. You can obtain the graph of h12 = b shifting the graph of ƒ12 =- two units to the right and five units up, as shown in Figure 2.19b. The -intercept of the graph of h is h102 = = =-11, which appears to agree with the transformed graph. Now tr Eercise 1.

2 186 CHAPTER 2 Polnomial, Power, and Rational Functions Eample 2 shows what can happen when simple monomial functions are combined to obtain polnomial functions. The resulting polnomials are not mere translations of monomials. EXAMPLE 2 Graphing Combinations of Monomial Functions [.7,.7] b [ 3.1, 3.1] [.7,.7] b [ 3.1, 3.1] FIGURE 2.20 The graph of ƒ12 = 3 + b itself and with =. (Eample 2a) Graph the polnomial function, locate its etrema and zeros, and eplain how it is related to the monomials from which it is built. ƒ12 = 3 + g12 = 3 - The graph of ƒ12 = 3 + is shown in Figure 2.20a. The function ƒ is increasing on 1-, 2, with no etrema. The function factors as ƒ12 = and has one zero at = 0. The general shape of the graph is much like the graph of its leading term 3, but near the origin ƒ behaves much like its other term, as shown in Figure 2.20b. The function ƒ is odd, just like its two building block monomials. The graph of g12 = 3 - is shown in Figure 2.21a. The function g has a local maimum of about 0.38 at and a local minimum of about at The function factors as g12 = and has zeros located at =-1, = 0, and = 1. The general shape of the graph is much like the graph of its leading term 3, but near the origin g behaves much like its other term -, as shown in Figure 2.21b. The function g is odd, just like its two building block monomials. Now tr Eercise 7. [.7,.7] b [ 3.1, 3.1] [.7,.7] b [ 3.1, 3.1] FIGURE 2.21 The graph of g12 = 3 - b itself and with =-. (Eample 2b) We have now seen a few eamples of graphs of polnomial functions, but are these tpical? What do graphs of polnomials look like in general? To begin our answer, let s first recall that ever polnomial function is defined and continuous for all real numbers. Not onl are graphs of polnomials unbroken without jumps or holes, but the are smooth, unbroken lines or curves, with no sharp corners or cusps. Tpical graphs of cubic and quartic functions are shown in Figures 2.22 and Imagine horizontal lines passing through the graphs in Figures 2.22 and 2.23, acting as -aes. Each intersection would be an -intercept that would correspond to a zero of the function. From this mental eperiment, we see that cubic functions have at most three zeros and quartic functions have at most four zeros. Focusing on the high and low points in Figures 2.22 and 2.23, we see that cubic functions have at most two local etrema and quartic functions have at most three local etrema. These observations generalize in the following wa:

3 SECTION 2.3 Polnomial Functions of Higher Degree with Modeling 187 a 3 > 0 a 3 < 0 FIGURE 2.22 Graphs of four tpical cubic functions: two with positive and two with negative leading coefficients. a > 0 a < 0 FIGURE 2.23 Graphs of four tpical quartic functions: two with positive and two with negative leading coefficients. THEOREM Local Etrema and Zeros of Polnomial Functions A polnomial function of degree n has at most n - 1 local etrema and at most n zeros. Technolog Note For a cubic, when ou change the horizontal window b a factor of k, it usuall is a good idea to change the vertical window b a factor of k 3. Similar statements can be made about polnomials of other degrees. End Behavior of Polnomial Functions An important characteristic of polnomial functions is their end behavior. As we shall see, the end behavior of a polnomial is closel related to the end behavior of its leading term. Eploration 1 eamines the end behavior of monomial functions, which are potential leading terms for polnomial functions. EXPLORATION 1 Investigating the End Behavior of ƒ12 = a n n Graph each function in the window 3-5, 5 b 3-15, 15. Describe the end behavior using lim ƒ12 and lim S S- ƒ ƒ12 = 2 3 ƒ12 =- 3 (c) ƒ12 = 5 (d) ƒ12 = ƒ12 =-3 ƒ12 = 0.6 (c) ƒ12 = 2 6 (d) ƒ12 = ƒ12 = ƒ12 =-2 2 (c) ƒ12 = 3 (d) ƒ12 = Describe the patterns ou observe. In particular, how do the values of the coefficient a n and the degree n affect the end behavior of ƒ12 = a n n?

4 188 CHAPTER 2 Polnomial, Power, and Rational Functions Eample 3 illustrates the link between the end behavior of a polnomial ƒ12 = a n n + g + a 1 + a 0 and its leading term a n n. EXAMPLE 3 Comparing the Graphs of a Polnomial and Its Leading Term [ 7, 7] b [ 25, 25] Superimpose the graphs of ƒ12 = and g12 = 3 in successivel larger viewing windows, a process called zoom out. Continue zooming out until the graphs look nearl identical. Figure 2.2 shows three views of the graphs of ƒ12 = and g12 = 3 in progressivel larger viewing windows. As the dimensions of the window increase, it gets harder to tell them apart. Moreover, lim ƒ12 = lim S S g12 = and lim S- ƒ12 = S- lim g12 =-. Now tr Eercise 13. [ 1, 1] b [ 200, 200] Eample 3 illustrates something that is true for all polnomials: In sufficientl large viewing windows, the graph of a polnomial and the graph of its leading term appear to be identical. Said another wa, the leading term dominates the behavior of the polnomial as S. Based on this fact and what we have seen in Eploration 1, there are four possible end behavior patterns for a polnomial function. The power and coefficient of the leading term tell us which one of the four patterns occurs. [ 56, 56] b [ 12800, 12800] (c) FIGURE 2.2 As the viewing window gets larger, the graphs of ƒ12 = and g12 = 3 look more and more alike. (Eample 3) Leading Term Test for Polnomial End Behavior For an polnomial function ƒ12 = a n n + g + a 1 + a 0, the limits lim ƒ12 and lim ƒ12 are determined b the degree n of the polnomial S S- and its leading coefficient a n : lim f() = lim f() = a n > 0 n odd a n < 0 n odd lim f() = lim f() = lim f() = lim f() = a n > 0 n even a n < 0 n even lim f() = lim f() =

5 SECTION 2.3 Polnomial Functions of Higher Degree with Modeling 189 EXAMPLE Appling Polnomial Theor Graph the polnomial in a window showing its etrema and zeros and its end behavior. Describe the end behavior using limits. ƒ12 = g12 = [ 5, 5] b [ 25, 25] The graph of ƒ12 = is shown in Figure 2.25a. The function ƒ has 2 etrema and 3 zeros, the maimum number possible for a cubic. lim S ƒ12 = and lim S- ƒ12 =-. The graph of g12 = is shown in Figure 2.25b. The function g has 3 etrema and zeros, the maimum number possible for a quartic. lim S g12 = and lim S- g12 =. Now tr Eercise 19. [ 5, 5] b [ 50, 50] FIGURE 2.25 Eample ƒ12 = , g12 = Zeros of Polnomial Functions Recall that finding the real number zeros of a function ƒ is equivalent to finding the -intercepts of the graph of = ƒ12 or the solutions to the equation ƒ12 = 0. Eample 5 illustrates that factoring a polnomial function makes solving these three related problems an eas matter. EXAMPLE 5 Finding the Zeros of a Polnomial Function Find the zeros of ƒ12 = Solve Algebraicall We solve the related equation ƒ12 = 0 b factoring: ( 2, 0) (0, 0) (3, 0) = = 0 Remove common factor = 0 = 0, - 3 = 0, or + 2 = 0 = 0, = 3, or =-2 So the zeros of ƒ are 0, 3, and -2. Factor quadratic. Zero factor propert Support Graphicall Use the features of our calculator to approimate the zeros of ƒ. Figure 2.26 shows that there are three values. Based on our algebraic solution we can be sure that these values are eact. Now tr Eercise 33. [ 5, 5] b [ 15, 15] FIGURE 2.26 The graph of = , showing the three -intercepts. (Eample 5) From Eample 5, we see that if a polnomial function ƒ is presented in factored form, each factor 1 - k2 corresponds to a zero = k, and if k is a real number, 1k, 02 is an -intercept of the graph of = ƒ12. When a factor is repeated, as in ƒ12 = , we sa the polnomial function has a repeated zero. The function ƒ has two repeated zeros. Because the factor - 2 occurs three times, 2 is a zero of multiplicit 3. Similarl, -1 is a zero of multiplicit 2. The following definition generalizes this concept. DEFINITION Multiplicit of a Zero of a Polnomial Function If ƒ is a polnomial function and 1 - c2 m is a factor of ƒ but 1 - c2 m+1 is not, then c is a zero of multiplicit m of ƒ.

6 190 CHAPTER 2 Polnomial, Power, and Rational Functions A zero of multiplicit m Ú 2 is a repeated zero. Notice in Figure 2.27 that the graph of ƒ just kisses the -ais without crossing it at 1-1, 02, but that the graph of ƒ crosses the -ais at 12, 02. This too can be generalized. [, ] b [ 10, 10] FIGURE 2.27 The graph of ƒ12 = , showing the -intercepts FIGURE 2.28 A sketch of the graph of ƒ12 = showing the -intercepts Zeros of Odd and Even Multiplicit If a polnomial function ƒ has a real zero c of odd multiplicit, then the graph of ƒ crosses the -ais at (c, 0) and the value of ƒ changes sign at = c. If a polnomial function ƒ has a real zero c of even multiplicit, then the graph of ƒ does not cross the -ais at (c, 0) and the value of ƒ does not change sign at = c. In Eample 5 none of the zeros were repeated. Because a nonrepeated zero has multiplicit 1, and 1 is odd, the graph of a polnomial function crosses the -ais and has a sign change at ever nonrepeated zero (Figure 2.26). Knowing where a graph crosses the -ais and where it doesn t is important in curve sketching and in solving inequalities. EXAMPLE 6 Sketching the Graph of a Factored Polnomial State the degree and list the zeros of the function ƒ12 = State the multiplicit of each zero and whether the graph crosses the -ais at the corresponding -intercept. Then sketch the graph of ƒ b hand. The degree of ƒ is 5 and the zeros are =-2 and = 1. The graph crosses the -ais at =-2 because the multiplicit 3 is odd. The graph does not cross at = 1 because the multiplicit 2 is even. Notice that values of ƒ are positive for 7 1, positive for , and negative for 6-2. Figure 2.28 shows a sketch of the graph of ƒ. Now tr Eercise 39. Intermediate Value Theorem The Intermediate Value Theorem tells us that a sign change implies a real zero. f 0 0 f FIGURE 2.29 If ƒ1a ƒ1b2 and ƒ is continuous on 3a, b, then there is a zero = c between a and b. a c b THEOREM Intermediate Value Theorem If a and b are real numbers with a 6 b and if f is continuous on the interval 3a, b, then f takes on ever value between ƒ1a2 and ƒ1b2. In other words, if 0 is between ƒ1a2 and ƒ1b2, then 0 = ƒ1c2 for some number c in 3a, b. In particular, if ƒ1a2 and ƒ1b2 have opposite signs (i.e., one is negative and the other is positive), then ƒ1c2 = 0 for some number c in 3a, b (Figure 2.29). EXAMPLE 7 Using the Intermediate Value Theorem Eplain wh a polnomial function of odd degree has at least one real zero. Let ƒ be a polnomial function of odd degree. Because ƒ is odd, the leading term test tells us that lim ƒ12 = -lim ƒ12. So there eist real numbers a S S- and b with a 6 b and such that ƒ and ƒ have opposite signs. Because ever polnomial function is defined and continuous for all real numbers, ƒ is continuous on the interval 3a, b. Therefore, b the Intermediate Value Theorem, ƒ(c) = 0 for some number c in 3a, b, and thus c is a real zero of ƒ. Now tr Eercise 61. In practice, the Intermediate Value Theorem is used in combination with our other mathematical knowledge and technological know-how.

7 SECTION 2.3 Polnomial Functions of Higher Degree with Modeling 191 Eact vs. Approimate In Eample 8, note that = 0.50 is an eact answer; the others are approimate. Use b-hand substitution to confirm that = 1/2 is an eact real zero. EXAMPLE 8 Zooming to Uncover Hidden Behavior Find all of the real zeros of ƒ12 = Solve Graphicall Because ƒ is of degree, there are at most four zeros. The graph in Figure 2.30a suggests a single zero (multiplicit 1) around =-3 and a triple zero (multiplicit 3) around = 1. Closer inspection around = 1 in Figure 2.30b reveals three separate zeros. Using the grapher, we find the four zeros to be 1.37, 1.13, = 0.50, and (See the margin note.) Now tr Eercise 75. [ 5, 5] b [ 50, 50] [0, 2] b [ 0.5, 0.5] FIGURE 2.30 Two views of ƒ12 = (Eample 8) X Y Y1 = X(20 2X)(25... FIGURE 2.32 A table to get a feel for the volume values in Eample 9. Modeling In the problem-solving process presented in Section 1.1, step 2 is to develop a mathematical model of the problem. When the model being developed is a polnomial function of higher degree, the algebraic and geometric thinking required can be rather involved. In solving Eample 9 ou ma find it helpful to make a phsical model out of paper or cardboard. EXAMPLE 9 Designing a Bo Diie Packaging Compan has contracted to make boes with a volume of approimatel 8 in. 3. Squares are to be cut from the corners of a 20-in. b 25-in. piece of cardboard, and the flaps folded up to make an open bo. (See Figure 2.31.) What size squares should be cut from the cardboard? Model We know that the volume V = height * length * width. So let 25 = edge of cut@out square (height of bo) 25-2 = length of the bo 20-2 = width of the bo V = FIGURE 2.31 [0, 10] b [0, 1000] FIGURE = and 2 = 8. (Eample 9) Solve Numericall and Graphicall For a volume of 8, we solve the equation = 8. Because the width of the cardboard is 20 in., We use the table in Figure 2.32 to get a sense of the volume values to set the window for the graph in Figure The cubic volume function intersects the constant volume of 8 at 1.22 and (continued)

8 192 CHAPTER 2 Polnomial, Power, and Rational Functions Interpret Squares with lengths of approimatel 1.22 in. or 6.87 in. should be cut from the cardboard to produce a bo with a volume of 8 in. 3. Now tr Eercise 67. Just as an two points in the Cartesian plane with different values and different values determine a unique slant line and its related linear function, an three noncollinear points with different values determine a quadratic function. In general, 1n + 12 points positioned with sufficient generalit determine a polnomial function of degree n. The process of fitting a polnomial of degree n to 1n + 12 points is polnomial interpolation. Eploration 2 involves two polnomial interpolation problems. EXPLORATION 2 Interpolating Points with a Polnomial 1. Use cubic regression to fit a curve through the four points given in the table Use quartic regression to fit a curve through the five points given in the table How good is the fit in each case? Wh? Generall we want a reason beond it fits well to choose a model for genuine data. However, when no theoretical basis eists for picking a model, a balance between goodness of fit and simplicit of model is sought. For polnomials, we tr to pick a model with the lowest possible degree that has a reasonabl good fit. Collected data usuall ehibit small-scale variabilit that is essentiall meaningless, so we should not tr to create a complicated curve that meanders through ever data point. Our objective is to find a simple curve that describes the underling relationship between the variables. QUICK REVIEW 2.3 (For help, go to Sections A.2. and P.5.) Eercise numbers with a gra background indicate problems that the authors have designed to be solved without a calculator. In Eercises 1 6, factor the polnomial into linear factors In Eercises 7 10, solve the equation mentall = = = ) ) 1-5) 3 = 0

9 SECTION 2.3 Polnomial Functions of Higher Degree with Modeling 193 SECTION 2.3 Eercises In Eercises 1 6, describe how to transform the graph of an appropriate monomial function ƒ12 = n into the graph of the given polnomial function. Sketch the transformed graph b hand and support our answer with a grapher. Compute the location of the -intercept as a check on the transformed graph. 1. g12 = g12 = g12 = g12 = g12 = g12 = In Eercises 7 and 8, graph the polnomial function, locate its etrema and zeros, and eplain how it is related to the monomials from which it is built. 7. ƒ12 = g12 = In Eercises 9 12, match the polnomial function with its graph. Eplain our choice. Do not use a graphing calculator. 20. ƒ12 = ƒ12 = ƒ12 = ƒ12 = ƒ12 = In Eercises 25 28, describe the end behavior of the polnomial function using lim ƒ12 and lim S S- ƒ ƒ12 = ƒ12 = ƒ12 = ƒ12 = In Eercises 29 32, match the polnomial function with its graph. Approimate all of the real zeros of the function. [ 5, 6] b [ 200, 00] [ 5, 6] b [ 200, 00] [, ] b [ 200, 200] [, ] b [ 200, 200] [ 5, 6] b [ 200, 00] (c) [ 5, 6] b [ 200, 00] (d) [ 2, 2] b [ 10, 50] (c) [, ] b [ 50, 50] (d) 9. ƒ12 = ƒ12 = ƒ12 = ƒ12 = In Eercises 13 16, graph the function pairs in the same series of viewing windows. Zoom out until the two graphs look nearl identical and state our final viewing window. 13. ƒ12 = and g12 = 3 1. ƒ12 = and g12 = ƒ12 = and g12 = ƒ12 = and g12 = 3 3 In Eercises 17 2, graph the function in a viewing window that shows all of its etrema and -intercepts. Describe the end behavior using limits. 17. ƒ12 = ƒ12 = ƒ12 = ƒ12 = ƒ12 = ƒ12 = ƒ12 = In Eercises 33 38, find the zeros of the function algebraicall. 33. ƒ12 = ƒ12 = ƒ12 = ƒ12 = ƒ12 = ƒ12 = In Eercises 39 2, state the degree and list the zeros of the polnomial function. State the multiplicit of each zero and whether the graph crosses the -ais at the corresponding -intercept. Then sketch the graph of the polnomial function b hand. 39. ƒ12 = ƒ12 = ƒ12 = ƒ12 =

10 19 CHAPTER 2 Polnomial, Power, and Rational Functions In Eercises 3 8, graph the function in a viewing window that shows all of its -intercepts and approimate all of its zeros. 3. ƒ12 = ƒ12 = ƒ12 = ƒ12 = ƒ12 = ƒ12 = In Eercises 9 52, find the zeros of the function algebraicall or graphicall. 9. ƒ12 = ƒ12 = ƒ12 = ƒ12 = In Eercises 53 56, using onl algebra, find a cubic function with the given zeros. Support b graphing our answer , -, , 3, , - 23, 56. 1, , Use cubic regression to fit a curve through the four points given in the table Use cubic regression to fit a curve through the four points given in the table Use quartic regression to fit a curve through the five points given in the table Use quartic regression to fit a curve through the five points given in the table (c) Superimpose the regression curve on the scatter plot. (d) Use the regression model to predict the stopping distance for a vehicle traveling at 25 mph. (e) For what speed would this regression model predict the stopping distance to be 300 ft? Table 2.1 Highwa Safet Division Speed (mph) Stopping Distance (ft) Analzing Profit Economists for Smith Brothers, Inc., find the compan profit P b using the formula P = R - C, where R is the total revenue generated b the business and C is the total cost of operating the business. Using data from past ears, the economists determined that R12 = models total revenue, and C12 = 12, models the total cost of doing business, where is the number of customers patronizing the business. For how man customers do these models predict that Smith Bros. would be profitable? For how man customers do these models predict that Smith Bros. would realize an annual profit of $60,000? 65. Circulation of Blood Research conducted at a national health research project shows that the speed at which a blood cell travels in an arter depends on its distance from the center of the arter. The function v = r 2 models the velocit (in centimeters per second) of a cell that is r centimeters from the center of an arter. r In Eercises 61 62, eplain wh the function has at least one real zero. 61. Writing to Learn ƒ12 = Writing to Learn ƒ12 = Stopping Distance A state highwa patrol safet division collected the data on stopping distances in Table 2.1 in the net column. Draw a scatter plot of the data. Find the quadratic regression model. Find a graph of v that reflects values of v appropriate for this problem. Record the viewing-window dimensions. At what distance from the center of the arter does this model predict that a blood cell travels at cm/sec? 66. Volume of a Bo Diie Packaging Co. has contracted to manufacture a bo with no top that is to be made b removing squares of width from the corners of a 15-in. b 60-in. piece of cardboard.

11 SECTION 2.3 Polnomial Functions of Higher Degree with Modeling 195 Show that the volume of the bo is modeled b V12 = Determine so that the volume of the bo is at least 50 in in. 15 in. 7. Multiple Choice Volume of a Bo Squares of width are removed from a 10-cm b 25-cm piece of cardboard, and the resulting edges are folded up to form a bo with no top. Determine all values of so that the volume of the resulting bo is at most 175 cm Volume of a Bo The function V = represents the volume of a bo that has been made b removing squares of width from each corner of a rectangular sheet of material and then folding up the sides. What values are possible for? Standardized Test Questions 69. True or False The graph of ƒ12 = crosses the -ais between = 1 and = 2. Justif our answer. 70. True or False If the graph of g12 = 1 + a2 2 is obtained b translating the graph of ƒ12 = 2 to the right, then a must be positive. Justif our answer. In Eercises 71 7, solve the problem without using a calculator. 71. Multiple Choice What is the -intercept of the graph of ƒ12 = ? (A) 7 (B) 5 (C) 3 (D) 2 (E) Multiple Choice What is the multiplicit of the zero = 2 in ƒ12 = ? (A) 1 (B) 2 (C) 3 (D) 5 (E) 7 In Eercises 73 and 7, which of the specified functions might have the given graph? 73. Multiple Choice (A) ƒ12 = (B) ƒ12 = (C) ƒ12 = (D) ƒ12 = (E) ƒ12 = Eplorations In Eercises 75 and 76, two views of the function are given. 75. Writing to Learn Describe wh each view of the function ƒ12 = , b itself, ma be considered inadequate. [ 5, 10] b [ 7500, 7500] [ 3, ] b [ 250, 100] 76. Writing to Learn Describe wh each view of the function ƒ12 = , b itself, ma be considered inadequate. 2 2 (A) ƒ12 = (B) ƒ12 = (C) ƒ12 = (D) ƒ12 = (E) ƒ12 = [ 6, ] b [ 2000, 2000] [0.5, 1.5] b [ 1, 1] In Eercises 77 80, the function has hidden behavior when viewed in the window 3-10, 10 b 3-10, 10. Describe what behavior is hidden, and state the dimensions of a viewing window that reveals the hidden behavior. 77. ƒ12 = ƒ12 = ƒ12 = ƒ12 =

12 196 CHAPTER 2 Polnomial, Power, and Rational Functions Etending the Ideas 81. Graph the left side of the equation = a1 - b2 3 + c. Then eplain wh there are no real numbers a, b, and c that make the equation true. (Hint: Use our knowledge of = 3 and transformations.) 82. Graph the left side of the equation = a1 - b2 + c. Then eplain wh there are no real numbers a, b, and c that make the equation true. 83. Looking Ahead to Calculus The figure shows a graph of both ƒ12 = and the line L defined b = (2, 7) [0, 5] b [ 10, 15] Confirm that the point Q12, 72 is a point of intersection of the two graphs. Zoom in at point Q to develop a visual understanding that = is a linear approimation for = ƒ12 near = 2. (c) Recall that a line is tangent to a circle at a point P if it intersects the circle onl at point P. View the two graphs in the window 3-5, 5 b 3-25, 25, and eplain wh that definition of tangent line is not valid for the graph of ƒ. 8. Looking Ahead to Calculus Consider the function ƒ12 = n where n is an odd integer. Suppose that a is a positive number. Show that the slope of the line through the points P1a, ƒ1a22 and Q1-a, ƒ1-a22 is a n-1. Let 0 = a 1/1n-12. Find an equation of the line through point 1 0, ƒ with the slope a n-1. (c) Consider the special case n = 3 and a = 3. Show both the graph of ƒ and the line from part b in the window 3-5, 5 b 3-30, Derive an Algebraic Model of a Problem Show that the distance in the figure is a solution of the equation ,000 = 0 and find the value of D b following these steps D u D 8 u Use the similar triangles in the diagram and the properties of proportions learned in geometr to show that 8 = Show that = - 8. (c) Show that 2-2 = 500. Then substitute for, and simplif to obtain the desired degree equation in. (d) Find the distance D. 86. Group Activit Consider functions of the form ƒ12 = 3 + b where b is a nonzero real number. Discuss as a group how the value of b affects the graph of the function. After completing, have each member of the group (individuall) predict what the graphs of ƒ12 = and g12 = will look like. (c) Compare our predictions with each other. Confirm whether the are correct.