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**Rational Functions and Models**

4.6 Identify a rational function and state its domain Identify asymptotes Interpret asymptotes Graph a rational function by using transformations Graph a rational function by hand Copyright © 2010 Pearson Education, Inc.

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**Rational Function A function f represented by**

where p(x) and q(x) are polynomials and q(x) ≠ 0, is a rational function. Copyright © 2010 Pearson Education, Inc.

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Rational Function The domain of a rational function includes all real numbers except the zeros of the denominator q(x). The graph of a rational function is continuous except at x-values where q(x) = 0. Copyright © 2010 Pearson Education, Inc.

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Example 1 For each rational function, determine any horizontal or vertical asymptotes. a) b) c) Copyright © 2010 Pearson Education, Inc.

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Example 1 Solution a) b) Is a rational function - both numerator and denominator are polynomials; domain is all real numbers; x2 + 1 ≠ 0 Is NOT a rational function Denominator is not a polynomial; domain is {x | x > 0} Copyright © 2010 Pearson Education, Inc.

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**Example 1 Solution continued c)**

Is a rational function - both numerator and denominator are polynomials; domain is {x | x ≠1, x ≠ 2} because (x – 1)(x – 2) = 0 when x = 1 and x = 2. Copyright © 2010 Pearson Education, Inc.

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Vertical Asymptotes The line x = k is a vertical asymptote of the graph of f if f(x) g ∞ or f(x) g –∞ as x approaches k from either the left or the right. Copyright © 2010 Pearson Education, Inc.

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**Horizontal Asymptotes**

The line y = b is a horizontal asymptote of the graph of f if f(x) g b as x approaches either ∞ or –∞. Copyright © 2010 Pearson Education, Inc.

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**Finding Vertical & Horizontal Asymptotes**

Let f be a rational function given by written in lowest terms. Vertical Asymptote To find a vertical asymptote, set the denominator, q(x), equal to 0 and solve. If k is a zero of q(x), then x = k is a vertical asymptote. Caution: If k is a zero of both q(x) and p(x), then f(x) is not written in lowest terms, and x – k is a common factor. Copyright © 2010 Pearson Education, Inc.

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**Finding Vertical & Horizontal Asymptotes**

(a) If the degree of the numerator is less than the degree of the denominator, then y = 0 (the x-axis) is a horizontal asymptote. (b) If the degree of the numerator equals the degree of the denominator, then y = a/b is a horizontal asymptote, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator. Copyright © 2010 Pearson Education, Inc.

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**Finding Vertical & Horizontal Asymptotes**

(c) If the degree of the numerator is greater than the degree of the denominator, then there are no horizontal asymptotes. Copyright © 2010 Pearson Education, Inc.

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Example 4 For each rational function, determine any horizontal or vertical asymptotes. Solution g(x) is a translation of f(x) left one unit and down 2 units. The vertical asymptote is x = 1 The horizontal asymptote is y = 2 g(x) = f(x + 1) 2 Copyright © 2010 Pearson Education, Inc.

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Example 4 For each rational function, determine any horizontal or vertical asymptotes. a) b) c) Copyright © 2010 Pearson Education, Inc.

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Example Solution a) Degree of numerator and denominator are both 1. Since the ratio of the leading coefficients is 6/3, the horizontal asymptote is y = 2. When x = –1, the denominator, 3x + 3, equals 0 and the numerator, 6x – 1 does not equal 0, so the vertical asymptote is x = 1 Copyright © 2010 Pearson Education, Inc.

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**Example Solution continued a) Here’s a graph of f(x).**

Copyright © 2010 Pearson Education, Inc.

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**Example Solution continued b)**

Degree of numerator is one less than the degree of the denominator so the x-axis, or y = 0, is a horizontal asymptote. When x = ±2, the denominator, x2 – 4, equals 0 and the numerator, x + 1 does not equal 0, so the vertical asymptotes are x = 2 and x = 2. Copyright © 2010 Pearson Education, Inc.

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**Example Solution continued b) Here’s a graph of g(x).**

Copyright © 2010 Pearson Education, Inc.

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Example Solution c) Degree of numerator is greater than the degree of the denominator so there are no horizontal asymptotes. When x = –1, both the numerator and denominator equal 0 so the expression is not in lowest terms: g(x) = x – 1, x ≠ –1. There are no vertical asymptotes. Copyright © 2010 Pearson Education, Inc.

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**Example Solution c) Here’s the graph of h(x). A straight line with the**

point (–1, –2) missing. Copyright © 2010 Pearson Education, Inc.

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**Slant, or Oblique, Asymptotes**

A third type of asymptote, which is neither vertical nor horizontal, occurs when the numerator of a rational function has degree one more than the degree of the denominator. Copyright © 2010 Pearson Education, Inc.

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**Slant, or Oblique, Asymptotes**

The line y = x + 1 is a slant asymptote, or oblique asymptote of the graph of f. Copyright © 2010 Pearson Education, Inc.

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**Graphs and Transformations of Rational Functions**

Graphs of rational functions can vary greatly in complexity. We begin by graphing and then use transformations to graph other rational functions. Copyright © 2010 Pearson Education, Inc.

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**Example 5 Sketch a graph of and identify any asymptotes. Solution**

Vertical asymptote: x = 0 Horizontal asymptote: y = 0 Copyright © 2010 Pearson Education, Inc.

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**Example 6 Use the graph of to sketch a graph of Include all asymptotes**

in your graph. Write g(x) in terms of f(x). Copyright © 2010 Pearson Education, Inc.

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Example 6 Solution g(x) is a translation of f(x) left 2 units and then a reflection across the x-axis. Vertical asymptote: x = –2 Horizontal asymptote: y = 0 g(x) = –f(x + 2) Copyright © 2010 Pearson Education, Inc.

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**Example 7 Let a) Use a calculator to graph f. Find the domain of f.**

b) Identify any vertical or horizontal asymptotes. c) Sketch a graph of f that includes the asymptotes. Copyright © 2010 Pearson Education, Inc.

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Example 7 Solution a) Here’s the calculator display using “Dot Mode.” The function is undefined when x2 – 4 = 0, or when x = ±2. The domain of f is D = {x|x ≠ 2, x ≠ –2}. Copyright © 2010 Pearson Education, Inc.

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**Example 7 Solution b) When x = ±2, the denominator x2 – 4**

= 0 (the numerator does not), so the vertical asymptotes are x = ±2. Degree of numerator = degree of denominator, ratio of leading coefficients is 2/1 = 2, so the horizontal asymptote is y = 2. Copyright © 2010 Pearson Education, Inc.

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**Example 7 Solution c) Here’s another version of the graph.**

Copyright © 2010 Pearson Education, Inc.

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**Graphing Rational Functions by Hand**

Let define a rational function in lowest terms. To sketch its graph, follow these steps. STEP 1: Find all vertical asymptotes. STEP 2: Find all horizontal or oblique asymptotes. STEP 3: Find the y-intercept, if possible, by evaluating f(0). Copyright © 2010 Pearson Education, Inc.

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**Graphing Rational Functions by Hand**

STEP 4: Find the x-intercepts, if any, by solving f(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 by solving f(x) = b, where b is the y-value of the horizontal asymptote, or by solving f(x) = mx + b, where y = mx + b is the equation of the oblique asymptote. Copyright © 2010 Pearson Education, Inc.

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**Graphing Rational Functions by Hand**

STEP 6: Plot selected points as necessary. Choose an x-value in each interval of the domain determined by the vertical asymptotes and x-intercepts. STEP 7: Complete the sketch. Copyright © 2010 Pearson Education, Inc.

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**Example 8 Graph Solution STEP 1: Vertical asymptote: x = 3**

STEP 2: Horizontal asymptote: y = 2 STEP 3: f(0) = , y-intercept is Copyright © 2010 Pearson Education, Inc.

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**Example 8 Solution continued STEP 4: Solve f(x) = 0 The x-intercept is**

Copyright © 2010 Pearson Education, Inc.

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**Example 8 Solution continued**

STEP 5: Graph does not intersect its horizontal asymptote, since f(x) = 2 has no solution. STEP 6: The points are on the graph. STEP 7: Complete the sketch (next slide) Copyright © 2010 Pearson Education, Inc.

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**Example 8 Solution continued STEP 7**

Copyright © 2010 Pearson Education, Inc.

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