11.4 Rational Functions and Their Graphs 411 The graph will look like one of the following. Figure 11.23 2. Find x-intercepts. Where is the numerator of the simplified expression zero? (For a fraction to be zero, its numerator must be zero.) 3. Positive/negative. Figure out the sign of f(x). The sign of any function can change only across a zero or a point of discontinuity. Draw a “sign of f(x)”number line. The sign can change only on either side of an x-intercept or vertical asymptote. Observe the following. If the factor (x − a) occurs in either numerator of f with odd multiplicity, then f changes sign around x = a. If it occurs with even multiplicity, then f does not change sign around x = a. This is consistent with the behavior we observed in polynomials. 4. Find the y-intercept. Set x = 0. (This merely gives a point of reference.) 5. Look for horizontal asymptotes. A horizontal asymptote includes the behavior of f(x) as x →±∞.Ahorizontal asymptote gives the behavior of f(x) only for x of very large magnitude; therefore, it can be crossed. Find lim x→∞ f(x). (For rational functions if lim x→∞ f(x)is finite it will be equal to lim x→−∞ f(x).) degree top > degree bottom ⇒ no horizontal asymptote |f(x)|→∞ degree top < degree bottom ⇒ horizontal asymptote at y = 0 (the x-axis) degree top = degree bottom ⇒ horizontal asymptote at y = fraction formed by leading coefficients 6. Consider symmetry. If the asymptotes and intercepts don’t have the required symme- try, then the function is neither even nor odd. If they do, then consider whether the function has even symmetry, odd symmetry, or neither. You can look at the numerator and denominator separately first. even = even even or odd odd ; odd = even odd or odd even 7. If you are interested in local extrema, then compute the derivative. The graphs of some rational functions can be built up from the sum or difference of other familiar rational functions. For example, h(x) = x 2 +1 x = x + 1 x . Refer to Figure 11.24 to see how the graph of x 2 +1 x can be built by summing the y-values of the graphs of x and 1/x. 412 CHAPTER 11 A Portrait of Polynomials and Rational Functions y y x x y x h(x) = = x + x 2 + 1 1 x x 1 x f(x) = x h(x) = f (x) + g(x) g(x) = vertical asymptote at x = 0 vertical asymptote at x = 0 f(x) h(x) g(x) Figure 11.24 EXERCISE 11.12 Differentiate h(x) in the example above to find the coordinates of the local extrema. Answer local minimum at x = 1, local maximum at x =−1. Below are worked examples; rational functions are presented and graphed. ◆ EXAMPLE 11.14 (a) f(x)= 4x(x+2) x−3 = 4x 2 +8x x−3 x-intercepts: x = 0 and x =−2 y-intercept: y = 0 vertical asymptotes: x = 3 symmetry? None. –230 – +–+sign of y horizontal asymptote? f(x)= 4x 2 +8x x−3 ,soasx→±∞,this looks like 4x 2 x = 4x. No horizontal asymptote. y x x = 3 –2 –1123 f(x) Figure 11.25 (b) f(x)= x 2 2x 2 −8 = x 2 2(x 2 −4) = x 2 2(x+2)(x−2) x-intercepts: x = 0 y-intercept: y = 0 vertical asymptotes: x = 2, x =−2 symmetry? Even, because f(−x) = f(x). Alternatively, even even = even. 11.4 Rational Functions and Their Graphs 413 –220 ++–– sign of y horizontal asymptote? y = x 2 2x 2 −8 ,soasx→±∞,this looks like y = x 2 2x 2 = 1 2 . Horizontal asymptote at y = 1 2 . y x x = 2 y = 1 /2 x = –2 f(x) Figure 11.26 (c) f(x)= (x+3)(x+1) 2 x 3 −3x 2 +2x = (x+3)(x+1) 2 x(x−2)(x−1) x-intercepts: x =−3, x =−1 y-intercept: None. (y-axis is asymptote) vertical asymptotes: x = 0, x = 2, x = 1 symmetry? None. –1–3210 (–)(–) (+)(–)(+)(+) sign of y horizontal asymptote? f(x)= (x−3)(x+1) 2 x 3 −3x 2 +2x ,soasx→±∞,this looks like x 3 x 3 = 1. Horizontal asymptote at y = 1. x y 12 3–3–2–1 x =1 y =1 x =2 x = 0 f(x) Figure 11.27 Question. Does the graph cross the line y = 1? (x+3)(x+1) 2 x 3 −3x 2 +2x = 1 or, multiplying out, x 3 +5x 2 +7x+3 x 3 −3x 2 +2x = 1. x 3 + 5x 2 + 7x + 3 = x 3 − 3x 2 + 2x,so5x 2 +7x+3=−3x 2 +2x, or 8x 2 + 5x + 3 = 0. The discriminant is negative, b 2 − 4ac = 25 −(4)(3)(8)<0,so there are no solutions. The graph does not cross the line y = 1. 414 CHAPTER 11 A Portrait of Polynomials and Rational Functions (d) f(x)= x−2 (x 2 +1)(x−2) . f(x)= 1 x 2 +1 for x = 2, undefined for x = 2 There is a pinhole at x = 2. There is just a pinhole because lim x→2 + 1 x 2 +1 = lim x→2 − 1 x 2 +1 = 1 5 . Both the left- and right-hand limits are equal and finite. x-intercepts: None y-intercept: 1 vertical asymptotes: None symmetry? If the point of discontinuity at x = 2 were removed, then f would be even. (+) sign of y horizontal asymptote? As x →±∞, f(x)= 1 x 2 +1 → 0. Horizontal asymptote at y = 0. 1 x 2 +1 is biggest when x 2 + 1 is smallest; the maximum value of 1 x 2 +1 is 1 and is attained at x = 0. y x f(x) Figure 11.28 REMARK If you merely follow a recipe for doing mathematics problems you will not always do them in the most efficient way possible. Example (d) is most easily done as follows: f(x)= x−2 (x 2 +1)(x−2) = 1 x 2 +1 for x = 2, and is undefined at x = 2. We can graph 1 x 2 +1 by beginning with the graph of g(x) = x 2 + 1 and then graphing its reciprocal, 1 g(x) . y x y x 1 1 y = y = x 2 +1 x 2 + 1 1 Figure 11.29 We insert a pinhole at x = 2 because f(x)is undefined at x = 2. 11.4 Rational Functions and Their Graphs 415 y x 2 Figure 11.30 ◆ Fitting a Function to a Graph We will use the graph below as reference. y x x = b ad x = c Figure 11.31 1. Find all x-intercepts x = a, x = d, and construct the factors (x − a), (x − d), for the numerator. Include a constant factor for stretching/flipping. Begin with y = k (x−a)(x−d) ? . 2. Find all vertical asymptotes x = b, x = c, and construct factors (x − b), (x − c), for the denominator. y = k (x − a)(x − d) (x − b)(x − c) 3. Do preliminary sign analysis. If the sign of y changes across an x-intercept or vertical asymptote at x = q, the factor (x − q) should be raised to an odd power; if the sign does not change, the factor must be raised to an even power. 4. Look at horizontal asymptotes. If there is no horizontal asymptote, be sure that (degree of the numerator) > (degree of the denominator). If the x-axis is a horizontal asymptote, be sure that (degree of the numerator) < (degree of the denominator). For any other horizontal asymptote, the degree of the numerator and denominator must be equal; adjusting k adjusts the height of the horizontal asymptote. Notice that introducing a factor like (x 2 + 1) in the numerator or denominator changes its degree without introducing new zeros or vertical asymptotes because (x 2 + 1) has no zeros. 416 CHAPTER 11 A Portrait of Polynomials and Rational Functions EXERCISE 11.13 Find functions that fit the graphs below. y x x = –1 x = 2 y = 2 (a) y x x = –1 x = 1 2 y = 3 (b) Figure 11.32 Possible answers (a) f(x)= 2x 3 (x+1)(x−2) 2 (b) f(x)= 3(x−2) 2 (x−1)(x+1) ◆ EXAMPLE 11.15 What are the absolute maximum and minimum values taken on by f(x)= x −2 x 2 + 5 ? SOLUTION First notice that f(x) has no vertical asymptotes, as x 2 + 5 > 0. Observe that lim x→∞ f(x)=0,so we do expect to find absolute extrema for this function. The extrema must occur at critical points of f .Wefind f using the Quotient Rule. f (x) = (x 2 + 5) · 1 − (x − 2) · 2x (x 2 + 5) 2 = x 2 + 5 − 2x 2 + 4x (x 2 + 5) 2 = −x 2 + 4x + 5 (x 2 + 5) 2 f (x) is always defined; we look for x such that f (x) = 0. −x 2 + 4x + 5 (x 2 + 5) 2 = 0 x 2 − 4x − 5 = 0 (x − 5)(x + 1) = 0 x = 5orx=−1 Critical points: x =−1and x = 5. 11.4 Rational Functions and Their Graphs 417 graph of f sign of f ′ (–) –15(+) (–) Figure 11.33 f (x) − −(x − 5)(x + 1) (x 2 + 5) 2 Minimum at x =−1. f(−1)= −1−2 (−1) 2 +5 = −3 6 =− 1 2 . This is the absolute minimum of f . Maximum at x = 5. f(5)= 5−2 25+5 = 3 30 = 1 10 . This is the absolute maximum value of f . ◆ PROBLEMS FOR SECTION 11.4 1. Find an equation to fit the important features of the graph. For the purposes of this problem, the important features are the horizontal and vertical asymptotes, x-intercepts, and the sign of y. y x y=2 y=2 y=1 x= –1 x= –1 y xx (a)(b) x= –1 y (c) –2 x x= –1 x=2 y (d) xx x=–1x=2 y (e) x= –3 x=1 y ( f) x x= –3 x=1 y (g) x x=0 y (h) 2 2. Suppose a distance function is given by d(t) = 1/t for 0.5 ≤ t ≤ 20. (a) What is the average velocity over the interval from t = 1tot=5? (b) Is there a time at which the instantaneous velocity is the same as the average velocity over the interval from t = 1tot=5? If so, find that time. (c) On the same set of axes, illustrate your answers to parts (a) and (b). 418 CHAPTER 11 A Portrait of Polynomials and Rational Functions 3. Find possible equations for the following graphs: y x x= –1 x=2 y=2 y=2 (a) y x –2 x= –1 (b) y x –2 x= –1 x=1 (c) 4. Let f(x)= x 2 +1 x 2 = 1 + 1 x 2 (a) Graph f . (b) Find the following. i. lim x→∞ f(x) ii. lim x→0 f(x) iii. lim x→∞ f (x) iv. lim x→0 + f (x) v. lim x→0 − f (x) (c) Graph f (x). (d) Find f (x). (e) Are your answers to all parts of this problem consistent? (If not, find your errors.) 5. Graph f(x)= x 3 +x 2 x 2 −4 . This function has four local extrema. One you can locate exactly. Where is it? Approximate the other three using your graphing calculator. (Notice that depending upon the viewing window you choose it may be very difficult to realize this function has four local extrema! When you use the calculator, use it carefully.) 6. Find a function whose graph matches the one drawn below. y y = 1 –2 x 7. Graph the following, clearly labeling all x- and y-intercepts, vertical asymptotes, and horizontal asymptotes. (a) y = x 2 −4 x 2 −3x−4 (b) y = 3x 2 (x−1) 2 (c) y = (x−1)(x−2) x(x−1)(x−3)(x+1) 11.4 Rational Functions and Their Graphs 419 8. Find possible equations to match with the following graphs. y y = – 4 y = –2 –33 x = –2 x = 1 x y x (a)(b) 9. At one point in Leo Tolstoy’s novella The Death of Ivan Ilyich, the title character states that the amount of blackness (the opposite of goodness in this context) in his life is in “inverse ratio to the square of the distance from death.” Let B(t) represent the amount of blackness in his life, where t measures the amount of time since his birth, and let t = D represent the time of his death. (a) Write an equation for B(t). (Your answer should include the constant D.) (b) Your equation for B(t) should have an arbitrary constant in it. Can you determine the sign of this constant? (c) Sketch a graph of B(t). Is it increasing or decreasing? Concave up or concave down? Label any t-orB-intercepts and any asymptotes. 10. What characteristics might the graph of a rational function (a polynomial divided by a polynomial) have that the graph of a polynomial will not have? 11. Graph f(x)= 4 x + x. (a) Find f (x). Make a number line, marking all points at which f is zero or unde- fined. Use the number line to indicate the sign of f ; above this indicate where the graph of f is increasing and where it is decreasing. Note: x = 0 is not a critical point, since f is undefined at x = 0. However, it is possible for the sign of f to change on either side of a point at which f is undefined, so x = 0 must be labeled on your number line. (b) Find f (x). Make a number line, marking all points at which f is zero and undefined. Use the number line to indicate the sign of f ; above this indicate where the graph of f is concave up and where it is concave down. (c) Graph f(x). Label both the x- and y-coordinates of the local maxima and local minima. (d) Does f(x)have an absolute maximum value? If so, what is it? Does f(x)have an absolute minimum value? If so, what is it? 12. Graph each of the following equations without using calculus. Label the following. (a) The x-intercepts; the y-intercepts (b) The vertical asymptotes (c) The horizontal asymptotes 420 CHAPTER 11 A Portrait of Polynomials and Rational Functions An analysis of where y is positive and where it is negative must be included. You need not find the coordinates of the local extrema. You need not look at y . i. y = x (x−1)(x+1) ii. y = x 2 (x−2) (x−1)(x+1) iii. y = x 2 (x−2) (x−1)(x+1) iv. y = x 2 (x−2) (x−1) 2 (x+1) v. y = x(x−2) (x−1) 2 (x+1) vi. y = (x−3)(x−2) (x−1)(x+1) vii. y = 2 x 2 +1 viii. y = −x 2 x 2 +1 13. Graph the following functions using the information provided by the derivatives for guidance. Indicate where the function is increasing, where it is decreasing, and the coordinates of all local extrema. (a) f(x)=x+ 1 x (b) g(x) = x − 1 x 14. Suppose that f(x) is a rational function with zeros at x = 0 and x = 4, vertical asymptotes at x =−2and x = 3, and a horizontal asymptote at y = 5. For each of the following functions, indicate the location of any (a) zeros. (b) vertical asymptotes. (c) horizontal asymptotes. If there is not enough information to answer part of any question, say so. i. g(x) = f(x−3) ii. h(x) = f(x)−3 iii. j(x)= 2f(3x) iv. k(x) = f(x 2 ) . = 0 and x = 4, vertical asymptotes at x =− 2and x = 3, and a horizontal asymptote at y = 5. For each of the following functions, indicate the location of any (a) zeros. (b) vertical asymptotes y-intercepts (b) The vertical asymptotes (c) The horizontal asymptotes 420 CHAPTER 11 A Portrait of Polynomials and Rational Functions An analysis of where y is positive and where it is negative must. interval from t = 1tot=5? If so, find that time. (c) On the same set of axes, illustrate your answers to parts (a) and (b). 418 CHAPTER 11 A Portrait of Polynomials and Rational Functions 3. Find