3.5. Rational Functions: Graphs, Applications, and Models
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1 3.5 Rational Functions: s, Applications, and Models The Reciprocal Function The Function Asympototes Steps for ing Rational Functions Rational Function Models Copyright 2008 Pearson Addison-Wesley. All rights reserved. 3-70
2 3.5 Example 1 ing a Rational Function (page 360). Give the domain and range. The expression can be written as, indicating that the graph can be obtained by stretching the graph of vertically by a factor of 4. Domain: Range: Copyright 2008 Pearson Addison-Wesley. All rights reserved. 3-71
3 3.5 Example 2 ing a Rational Function (page 361). Give the domain and range. Shift the graph of three units to the right, then reflect the graph across the x-axis to obtain the graph of. The horizontal asymptote is y = 0. The vertical asymptote is x = 3. Domain: Range: Copyright 2008 Pearson Addison-Wesley. All rights reserved. 3-72
4 3.5 Example 3 ing a Rational Function (page 363). Give the domain and range. To obtain the graph shift the graph of two units to the right, stretch the graph vertically by a factor of two, and then shift the graph four units down. The vertical asymptote is x = 2. The horizontal asymptote is y = 4. Domain: Range: Copyright 2008 Pearson Addison-Wesley. All rights reserved. 3-73
5 3.5 Example 4(a) Finding Asymptotes of Rational Functions (page 364) Find all asymptotes of the function To find the vertical asymptotes, set the denominator equal to 0 and solve. The equations of the vertical asymptotes are x = 4 and x = 4. The denominator has a bigger degree than the numerator, so the horizontal asymptote is y = 0. Copyright 2008 Pearson Addison-Wesley. All rights reserved. 3-74
6 3.5 Example 4(b) Finding Asymptotes of Rational Functions (page 364) Find all asymptotes of the function To find the vertical asymptotes, set the denominator equal to 0 and solve. The equation of the vertical asymptote is. The numerator and the denominator have the equal degrees, so the equation of the horizontal asymptote is Copyright 2008 Pearson Addison-Wesley. All rights reserved. 3-75
7 3.5 Example 4(c) Finding Asymptotes of Rational Functions (page 364) Find all asymptotes of the function To find the vertical asymptotes, set the den = 0 and solve. Vert asy is x = 3. Since the degree of the numerator is exactly one more than the denominator, there is an oblique asymptote. Divide the numerator by the denominator and, disregarding the remainder, set the rest of the quotient equal to y to obtain the equation of the asymptote. Oblique asy is y = 2x
8 3.5 Example 5 ing a Rational Function with the x-axis as Horizontal Asymptote (page 366) To find the vertical asymptotes, set the den = 0 and solve. Find the horizontal asymptote. The degree of den > degree of num, so the horizontal asy is y = 0. Find the y-intercept Find the x-intercept 3-77
9 3.5 Example 6 ing a Rational Function That Does Not Intersect its Horizontal Asymptote (page 367) To find the vertical asymptotes, set the den = 0 and solve. Find the horizontal asymptote. The degree of num = degree of den, so the horizontal asymptote is y = 4. Find the y-int Find the x-int 3-78
10 3.5 Ex 7 ing a Rational Function That Intersects its Horizontal Asymptote (page 368) Find the vertical asy Find the x-int Calculus bound students only Find the horizontal asy. Degree of num = degree of denom, so the horizontal asy is y = 2. Find the y-int
11 3.5 Example 8 ing a Rational Function With an Oblique Asymptote (page 369) Find the y-int Find the x-int Find the vertical asy Find the horizontal asy. Since the deg num > deg den there isn t a horiz asy. Divide to find oblique asy. x(x+1) = 0 x = 0 or -1 oblique asy is y = x + 2
12 3.5 Example 9 ing a Rational Function Defined by an Expression That is not in Lowest Terms (page 370) The domain of f cannot include 3. In lowest terms, the function becomes The graph of f is the same as the graph of y = x + 2, with the exception of the point with x-value 3. A hole appears in the graph at (3, 5). Copyright 2008 Pearson Addison-Wesley. All rights reserved. 3-81
13 3.5 Example 9 ing a Rational Function Defined by an Expression That is not in Lowest Terms (cont.) If the window of a graphing calculator is set so that an x-value of 3 is displayed, then the calculator cannot determine a value for y. Notice the visible discontinuity at x = 3. The error message in the table supports the existence of the discontinuity. Copyright 2008 Pearson Addison-Wesley. All rights reserved. 3-82
14 3.5 Summary Rational s X-intercept: solve for x, when y = 0 Y-intercept: solve for y, when x = 0 Vertical asymptote: set the denominator = 0 Horizontal asymptote: degree of num = degree of den then y = coef (BETC) degree of den > degree of num then y = 0 (BOBO) Oblique asymptote: degree of num > degree of den then y = quotient Holes occur in the graph when a factor reduces from the numerator and denominator Copyright 2008 Pearson Addison-Wesley. All rights reserved. 3-83
3.5. Rational Functions: Graphs, Applications, and Models. 3.5 Rational Functions: Graphs, Applications, and Models 3.6 Variation
3 Polynomial and Rational Functions 3 Polynomial and Rational Functions 3.5 Rational Functions: s, Applications, and Models 3.6 Variation Sections 3.5 3.6 2008 Pearson Addison-Wesley. All rights reserved
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