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COLLEGE ALGEBRA

COLLEGE ALGEBRA. LIAL HORNSBY SCHNEIDER. 3.5. Rational Functions: Graphs, Applications, and Models. The Reciprocal Function The Function Asymptotes Steps for Graphing Rational Functions Rational Function Models. Rational function. A function  of the form.

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COLLEGE ALGEBRA

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  1. COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER

  2. 3.5 Rational Functions: Graphs, Applications, and Models The Reciprocal Function The Function Asymptotes Steps for Graphing Rational Functions Rational Function Models

  3. Rational function A function  of the form where p(x) and q(x) are polynomials, with q(x) ≠ 0, is called a rational function.

  4. Rational Function Some examples of rational functions are Since any values of x such that q(x) = 0 are excluded from the domain of a rational function, this type of function often has a discontinuous graph, that is, a graph that has one or more breaks in it.

  5. The Reciprocal Function The simplest rational function with a variable denominator is the reciprocal function, defined by

  6. The Reciprocal Function The domain of this function is the set of all real numbers except 0. The number 0 cannot be used as a value of x, but it is helpful to find values of for some values of (x) for some values of x very close to 0. We use the table feature of a graphing calculator to do this. The tables suggest that (x) gets larger and larger as x gets closer and closer to 0, which is written in symbols as

  7. The Reciprocal Function (The symbol x  0 means that x approaches 0, without necessarily ever being equal to 0.) Since x cannot equal 0, the graph of will never intersect the vertical line x = 0.This line is called a vertical asymptote. On the other hand, as x gets larger and larger, the values of get closer and closer to 0, as shown in the tables. Letting x get larger and larger without bound (written x ) causes the graph to move closer and closer to the horizontal line. This line is called a horizontal asymptote.

  8. Domain: (–, 0)  (0, ) Range: (–, 0)  (0, ) RECIPROCAL FUNCTION • decreases on the intervals (–,0) and (0, ).

  9. Domain: (–, 0)  (0, ) Range: (–, 0)  (0, ) RECIPROCAL FUNCTION • It is discontinuous at x = 0.

  10. Domain: (–, 0)  (0, ) Range: (–, 0)  (0, ) RECIPROCAL FUNCTION • The y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote.

  11. Domain: (–, 0)  (0, ) Range: (–, 0)  (0, ) RECIPROCAL FUNCTION • It is an odd function, and its graph is symmetric with respect to the origin.

  12. GRAPHING A RATIONAL FUNCTION Example 1 Graph Give the domain and range. Solution The expression can be written as or indicating that the graph may be obtained by stretching the graph of vertically by a factor of 2 and reflecting it across either the y-axis or x-axis. The x- and y-axes remain the horizontal and vertical asymptotes. The domain and range are both still (–, 0)  (0, ).

  13. GRAPHING A RATIONAL FUNCTION Example 2 Graph Give the domain and range. Solution The expression can be written as indicating that the graph may be obtained by shifting the graph of to the left 1 unit and stretching it vertically by a factor of 2.

  14. GRAPHING A RATIONAL FUNCTION Example 2 Graph Give the domain and range. Solution The horizontal shift affects the domain, which is now (–, –1)  (–1, ) . The line x = –1 is the vertical asymptote, and the line y = 0 (the x-axis) remains the horizontal asymptote. The range is still (–, 0)  (0, ).

  15. The Function The Function The rational function defined by also has domain (–, 0)  (0, ). We can use the table feature of a graphing calculator to examine values of (x) for some x-values close to 0.

  16. The Function The tables suggest that (x) gets larger and larger as x gets closer and closer to 0. Notice that as x approaches 0 from either side, function values are all positive and there is symmetry with respect to the y-axis. Thus, (x)   as x  0. The y-axis (x = 0)is the vertical asymptote.

  17. Domain: (–, 0)  (0, ) Range: (0, ) RECIPROCAL FUNCTION • increases on the interval (–,0) and decreases on the interval (0, ).

  18. Domain: (–, 0)  (0, ) Range: (0, ) RECIPROCAL FUNCTION • It is discontinuous at x = 0.

  19. Domain: (–, 0)  (0, ) Range: (0, ) RECIPROCAL FUNCTION • The y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote.

  20. Domain: (–, 0)  (0, ) Range: (0, ) RECIPROCAL FUNCTION • It is an even function, and its graph is symmetric with respect to the y-axis.

  21. GRAPHING A RATIONAL FUNCTION Example 3 Graph Give the domain and range. Solution The equation is equivalent to where This indicates that the graph will be shifted 2 units to the left and 1 unit down.

  22. GRAPHING A RATIONAL FUNCTION Example 3 Graph Give the domain and range. Solution The equation is equivalent to The horizontal shift affects the domain, which is now (–, –2)  (–2, ), while the vertical shift affects the range, now (–1, ).

  23. GRAPHING A RATIONAL FUNCTION Example 3 Graph Give the domain and range. Solution The equation is equivalent to The vertical asymptote has equation x = –2, and the horizontal asymptote has equation y = –1.

  24. Asymptotes Let p(x) and q(x) define polynomials. For the rational function defined by written in lowest terms, and for real numbers a and b: 1. If (x)   as x  a, then the line x = a is a vertical asymptote. 2. If (x) b asx  , then the line y = b is a horizontal asymptote.

  25. Determining Asymptotes To find the asymptotes of a rational function defined by a rational expression in lowest terms, use the following procedures. 1. Vertical Asymptotes Find any vertical asymptotes by setting the denominator equal to 0 and solving for x. If a is a zero of the denominator, then the line x = a is a vertical asymptote.

  26. Determining Asymptotes 2. Other Asymptotes Determine any other asymptotes. Consider three possibilities: (a) If the numerator has lower degree than the denominator, then there is a horizontal asymptote y = 0 (the x-axis).

  27. Determining Asymptotes 2. Other Asymptotes Determine any other asymptotes. Consider three possibilities: (b) If the numerator and denominator have the same degree, and the function is of the form where an, bn ≠ 0, then the horizontal asymptote has equation

  28. Determining Asymptotes 2. Other Asymptotes Determine any other asymptotes. Consider three possibilities: (c) If the numerator is of degree exactly one more than the denominator, then there will be an oblique (slanted) asymptote. To find it, divide the numerator by the denominator and disregard the remainder. Set the rest of the quotient equal to y to obtain the equation of the asymptote.

  29. Motion Problems Note The graph of a rational function may have more than one vertical asymptote, or it may have none at all. The graph cannot intersect any vertical asymptote. There can be at most one other (nonvertical) asymptote, and the graph may intersect that asymptote as we shall see in Example 7.

  30. FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS Example 4 For each rational function , find all asymptotes. a. SolutionTo find the vertical asymptotes, set the denominator equal to 0 and solve. Zero-property Solve each equation.

  31. FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS Example 4 The equations of the vertical asymptotes are x = ½ and x = –3. To find the equation of the horizontal asymptote, divide each term by the greatest power of x in the expression. First, multiply the factors in the denominator.

  32. FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS Example 4 Now divide each term in the numerator and denominator by x2 since 2 is the greatest power of x. Stop here. Leave the expression in complex form.

  33. FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS Example 4 As x gets larger and larger, the quotients all approach 0, and the value of (x) approaches The line y = 0 (that is, the x-axis) is therefore the horizontal asymptote.

  34. FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS Example 4 For each rational function , find all asymptotes. b. SolutionSet the denominator x – 3 = 0 equal to 0 to find that the vertical asymptote has equation x = 3. To find the horizontal asymptote, divide each term in the rational expression by x since the greatest power of x in the expression is 1.

  35. FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS Example 4 For each rational function , find all asymptotes. b. Solution

  36. FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS Example 4 As x gets larger and larger, both approach 0, and (x) approaches so the line y = 2 is the horizontal asymptote.

  37. FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS Example 4 For each rational function , find all asymptotes. c. Solution Setting the denominator x – 2 equal to 0 shows that the vertical asymptote has equation x = 2. If we divide by the greatest power of x as before ( in this case), we see that there is no horizontal asymptote because

  38. FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS Example 4 does not approach any real number as x  , since is undefined. This happens whenever the degree of the numerator is greater than the degree of the denominator. In such cases, divide the denominator into the numerator to write the expression in another form. We use synthetic division.

  39. FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS Example 4 We use synthetic division. The result allows us to write the function as

  40. FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS Example 4 For very large values of x, is close to 0, and the graph approaches the line y = x + 2. This line is an oblique asymptote (slanted, neither vertical nor horizontal) for the graph of the function.

  41. Steps for Graphing Functions A comprehensive graph of a rational function exhibits these features: 1. all x-and y-intercepts; 2. all asymptotes: vertical, horizontal, and/or oblique; 3. the point at which the graph intersects its nonvertical asymptote (if there is any such point); 4. the behavior of the function on each domain interval determined by the vertical asymptotes and x-intercepts.

  42. Graphing a Rational Function Let define a function where p(x) and q(x) are polynomials and the rational expression is written in lowest terms. To sketch its graph, follow these steps. Step 1 Find any vertical asymptotes. Step 2 Find any horizontal or oblique asymptotes. Step 3 Find the y-intercept by evaluating (0).

  43. Graphing a Rational Function Step 4 Find the x-intercepts, if any, by solving (x) = 0 . (These will be the zeros of the numerator, p(x).) Step 5 Determine whether the graph will intersect its nonvertical asymptote y = b or y = mx + b by solving (x) = b or(x) = mx + b.

  44. Graphing a Rational Function Step 6 Plot selected points, as necessary. Choose an x-value in each domain interval determined by the vertical asymptotes and x-intercepts. Step 7 Complete the sketch.

  45. GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE Example 5 Graph Solution Step 1 Since 2x2 + 5x – 3 = (2x – 1)(x + 3), from Example 4(a), the vertical asymptotes have equations x = ½ and x = –3. Step 2 Again, as shown in Example 4(a), the horizontal asymptote is the x-axis.

  46. GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE Example 5 Graph Solution Step 3 The y-intercept is –⅓, since The y-intercept is the ratio of the constant terms.

  47. GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE Example 5 Graph Solution Step 4 The x-intercept is found by solving (x) = 0. If a rational expression is equal to 0, then its numerator must equal 0. The x-intercept is –1.

  48. GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE Example 5 Graph Solution Step 5 To determine whether the graph intersects its horizontal asymptote, solve y-value of horizontal asymptote Since the horizontal asymptote is the x-axis, the solution of this equation was found in Step 4. The graph intersects its horizontal asymptote at (–1, 0).

  49. GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE Example 5 Graph Solution Step 6 Plot a point in each of the intervals determined by the x-intercepts and vertical asymptotes, to get an idea of how the graph behaves in each interval.

  50. GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE Example 5

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