1. Introduction The graphs of rational functions can be sketched by knowing how to calculate horizontal and/or vertical asymptotes of the function and its zeros. The sketch of the graph can be improved by plotting points with x- and y-values that are at or near the limit of those domain and range values. Graphing calculators can be used to sketch the graph of a rational function, and can also provide the user with a table of x- and y-values from which these critical features of the graph can be read. 1 3.3.2: Graphing Rational Functions

2. Key Concepts There are several key features of the graph of a rational function that are helpful when creating a sketch, including its zeros, horizontal and vertical asymptotes, domain, range, and extreme values. The range is the set of all outputs of a function; that is, the set of y-values that are valid for the function. Recall that the zeros of the rational function are the domain value(s) for which f(x) = 0. If the rational function f(x) = 0 is represented by, then g(x) = 0 and h(x) ≠ 0. 2 3.3.2: Graphing Rational Functions

3. Key Concepts, continued If a rational function f(x) = 0 is represented by, then (x – a)(x – b) = 0 and (x – c)(x – d) ≠ 0. The zeros are represented by the points (a, 0), (b, 0), etc., on the graph. 3 3.3.2: Graphing Rational Functions

4. Key Concepts, continued Asymptotes are straight lines on the graph of a rational function that form the boundaries for parts of the graph. In some cases, the domain values of the asymptotes are the domain values for which the rational function is undefined. In other cases, the domain values of the asymptotes form the maximum and/or minimum value(s) of the rational function. 4 3.3.2: Graphing Rational Functions

5. Key Concepts, continued Vertical asymptotes are straight lines of the form x = a. For example, the graph of the rational function has vertical asymptotes at x = ±2, which are solutions to the equation x 2 – 4 = 0. The function f(x) is undefined at these domain values. 5 3.3.2: Graphing Rational Functions

6. Key Concepts, continued Horizontal asymptotes are straight lines of the form y = b. For example, the graph of the function has a horizontal asymptote at y = 4. As seen in the following graph, g(x) approaches y = 4. 6 3.3.2: Graphing Rational Functions

7. Key Concepts, continued 7 3.3.2: Graphing Rational Functions

8. Key Concepts, continued Slant asymptotes are asymptotes of a graph of the form y = ax + b, where a ≠ 0. A rational function that has a numerator with a degree that is exactly 1 more than the degree of the denominator will have a slant asymptote. 8 3.3.2: Graphing Rational Functions

9. Key Concepts, continued 9 3.3.2: Graphing Rational Functions

10. Key Concepts, continued To find the slant asymptote, use polynomial long division. The result of dividing the numerator by the denominator is the equation of the slant asymptote. Rational functions may or may not have a vertical asymptote, but will always have either a horizontal or a slant asymptote. A rational function will never have both a horizontal asymptote and a slant asymptote. 10 3.3.2: Graphing Rational Functions

11. Key Concepts, continued For the graph of a rational function, it can be useful to plot several points that represent values of x and y that are far away from zeros and asymptotes. Such a value is known as an extreme value—a point on the graph of a rational function that occurs at or near the endpoints of its domain or range. These so-called extreme values will ensure that the sketch will be accurate over the domain and range of the function. 11 3.3.2: Graphing Rational Functions

12. Key Concepts, continued For example, consider the function. The domain of the function is all numbers except for x = 1 since the function is undefined for this value of x. For values of x very close to x = 1, the value of the function is large. For x = 1.01, f(x) = 101, and for x = 0.99, f(x) = 99. But, for x = 100, f(x) = 1.1, and for x = –100, f(x) = 0.99. Sketching the points (1.01, 101), (0.99, 99), (100, 1.1), and (–100, 0.99) shows that f(x) has a horizontal asymptote at y = 1 and a vertical asymptote at x = 1. 12 3.3.2: Graphing Rational Functions

13. Key Concepts, continued 13 3.3.2: Graphing Rational Functions

14. Key Concepts, continued An example of a function without a vertical asymptote is because there are no values of x that will result in a denominator that equals 0. However, g(x) does have horizontal asymptotes at x = ±1. The extreme values of g(x) occur at x = ±1: and The extreme values of x occur over the domain (−∞,∞). If x = 100, g(100) = 0.06; if x = –100, g(–100) = –0.06. In other words, the graph of g(x) approaches 0 as x approaches ±100. 14 3.3.2: Graphing Rational Functions

15. Key Concepts, continued These ideas can be explored and extended using a graphing calculator. Refer to the previous sub-lesson for directions specific to your calculator model. 15 3.3.2: Graphing Rational Functions

16. Common Errors/Misconceptions identifying the factors of the numerator of a rational function as resulting in the function being undefined at those domain values identifying solutions for the denominator of a rational function that are not in the function’s domain writing the horizontal asymptote(s) of a rational function in the form x = a 16 3.3.2: Graphing Rational Functions

17. Common Errors/Misconceptions, continued writing the vertical asymptote(s) of a rational function in the form y = b identifying the extreme values of a rational function’s domain as being the extreme values of its range and vice versa 17 3.3.2: Graphing Rational Functions

18. Guided Practice Example 1 Sketch the graph of the rational function. Include points on both sides of any asymptotes. 18 3.3.2: Graphing Rational Functions

19. Guided Practice: Example 1, continued 1.Find the value of f(0). The point for which x = 0 is the y-intercept of the graph, and has the form (0, f(0)). The function value f(0) may or may not be equal to 0. The point on the graph representing f(0) is (0, 1). 19 3.3.2: Graphing Rational Functions

20. Guided Practice: Example 1, continued 2.Find the value(s) of x for which f(x) = 0. The values of x for which f(x) = 0 are the zeros of the function. If f(x) = 0, x 2 + 2 ≠ 0, so 2 – x 2 = 0, which means that. Therefore, the zeros of f(x) are given by and. 20 3.3.2: Graphing Rational Functions

21. Guided Practice: Example 1, continued 3.Write the equation(s) for the vertical asymptote(s) of f(x), if any. Vertical asymptotes of f(x), if any, occur at values of x which make the denominator equal 0. In this case, setting the denominator equal to 0 and solving for x 2 + 2 = 0 will not yield values of x that are within the domain of the function, which is (−∞,∞). Therefore, there are no vertical asymptotes for f(x). 21 3.3.2: Graphing Rational Functions

22. Guided Practice: Example 1, continued 4.Write the equation(s) of the horizontal asymptote(s) of f(x), if any. The horizontal asymptotes are of the form y = b. One way to find b is to solve the rational function for the domain variable, x. 22 3.3.2: Graphing Rational Functions Original function Rewrite the function as two fractions.

23. Guided Practice: Example 1, continued 23 3.3.2: Graphing Rational Functions f(x) (x 2 + 2) = 1 (2 – x 2 )Cross multiply. x 2 f(x) + 2 f(x) = 2 – x 2 Simplify. x 2 f(x) + x 2 = 2 – 2 f(x) Use subtraction and addition to group like terms (x 2 and 2 f(x)) on the same sides of the equation. x 2 [f(x) + 1] = 2 [1 – f(x)]Simplify.

24. Guided Practice: Example 1, continued This rational expression for x is undefined when the denominator f(x) + 1 = 0, so the horizontal asymptote is y = –1. 24 3.3.2: Graphing Rational Functions Divide. Solve for x by taking the square root of both sides.

25. Guided Practice: Example 1, continued 5.To prepare for sketching the graph of f(x), pick values for x that meet the conditions for f(x) = 0 from step 1, along with some values of,, and Then, calculate the value of f(x) for each value of x. List the calculations in a table. 25 3.3.2: Graphing Rational Functions

26. Guided Practice: Example 1, continued Let x = –2, –1, 0, 1, 2, and 3 and solve for the corresponding f(x) value in the original function. 26 3.3.2: Graphing Rational Functions x–2–10123 f(x)f(x)–0.330.331 –0.33–0.64

27. Guided Practice: Example 1, continued 6.List the points calculated in steps 1 and 2 for the y-intercepts and zeros of the function in a table. 27 3.3.2: Graphing Rational Functions x0 f(x)f(x)010

28. Guided Practice: Example 1, continued 7.Use your results to sketch the graph of the function. To sketch the graph, first plot the points listed in the tables in steps 5 and 6. The conditions listed in steps 5 and 6 will ensure that all of the regions on the coordinate plane that contain parts of the rational function’s graph will be represented. 28 3.3.2: Graphing Rational Functions

29. Guided Practice: Example 1, continued Use dotted lines to sketch the horizontal asymptote, y = –1. Use a graphing calculator to check your work. Refer to the appropriate calculator directions for your calculator model as provided in the Key Concepts in the previous sub-lesson. 29 3.3.2: Graphing Rational Functions

30. Guided Practice: Example 1, continued Your graph should resemble the following: 30 3.3.2: Graphing Rational Functions ✔

31. Guided Practice: Example 1, continued 31 3.3.2: Graphing Rational Functions

32. Guided Practice Example 2 Sketch the graph of the rational function, which represents the sum of a number and its reciprocal. 32 3.3.2: Graphing Rational Functions

33. Guided Practice: Example 2, continued 1.Find the value of f(0). The point for which x = 0 is the y-intercept of the graph, and has the form (0, f(0))., which is undefined. There is no point on the graph representing f(0). Therefore, x = 0 is a vertical asymptote. 33 3.3.2: Graphing Rational Functions

34. Guided Practice: Example 2, continued 2.Find the value(s) of x for which f(x) = 0. The values of x for which f(x) = 0 are the zeros of the function. Rewrite the function as one fraction. 34 3.3.2: Graphing Rational Functions Original function Write the term x as a fraction.

35. Guided Practice: Example 2, continued 35 3.3.2: Graphing Rational Functions Multiply by (which is equivalent to 1). Multiply. Simplify.

36. Guided Practice: Example 2, continued If f(0) = 0, then x 2 + 1 = 0 (assuming that x ≠ 0 from step 1). The equation x 2 + 1 = 0 has no solution on the interval (−∞,∞), so this rational function has no zeros. 36 3.3.2: Graphing Rational Functions

37. Guided Practice: Example 2, continued 3.Find any other asymptote(s) by solving the function for x. Rearrange and solve the function for x to reveal what values of f(x) are possible for the function. This will also indicate any values that the function cannot have. 37 3.3.2: Graphing Rational Functions

38. Guided Practice: Example 2, continued 38 3.3.2: Graphing Rational Functions Function from the previous step Rewrite the function as two fractions. f(x) (x) = 1 (x 2 + 1)Cross multiply. x[f(x)] = x 2 + 1Simplify. 0 = x 2 – x[f(x)] + 1 Use subtraction to collect all like terms on one side of the equation.

39. Guided Practice: Example 2, continued Two values of f(x) that allow this quadratic equation to be solved are f(x) = ±2. Values of f(x) ≥ 2 and f(x) ≤ –2 result in domain values (values of x) that are in the domain of the function. However, values of f(x) such that –2 < f(x) < 2 cannot be achieved with the domain values of the function. Notice that the function has a numerator that is exactly one degree higher than that of the denominator. This indicates a slant asymptote. 39 3.3.2: Graphing Rational Functions

40. Guided Practice: Example 2, continued To determine the equation of the slant asymptote, use polynomial long division to divide x 2 + 1 by x. The slant asymptote is the polynomial part of the answer, x, not the remainder. Therefore, the slant asymptote is y = x. 40 3.3.2: Graphing Rational Functions

41. Guided Practice: Example 2, continued 4.Use the results of steps 1–3 to create a table of at least 6 points that can be used to sketch the graph. Recall from step 1 that there is a vertical asymptote at x = 0. There also is a gap in the range of function values between f(x) = 2 and f(x) = –2. Points on either side of these range values should begin to suggest the shape of the graph. 41 3.3.2: Graphing Rational Functions

42. Guided Practice: Example 2, continued Therefore, choose values of x that result in range values close to these function values and solve for f(x). Summarize your results in a table. 42 3.3.2: Graphing Rational Functions x–3–2–10123 f(x)f(x)–3.33–2.5–2—22.53.33

43. Guided Practice: Example 2, continued 5.Use your results to sketch the graph of the function. To sketch the graph, first plot the points listed in the table in step 4. Use a dotted line to sketch the vertical asymptote, x = 0, and the slant asymptote, y = x. Use a graphing calculator to check your work. Refer to the appropriate calculator directions for your calculator model as provided in the Key Concepts in the previous sub-lesson. 43 3.3.2: Graphing Rational Functions

44. Guided Practice: Example 2, continued Your graph should resemble the following: 44 3.3.2: Graphing Rational Functions ✔

45. Guided Practice: Example 2, continued 45 3.3.2: Graphing Rational Functions