Explanation Approach: Let f ( x ) = p ( x ) q ( x ) define a rational function where p ( x ) and q ( x ) are polynomial functions and f ( x ) is written in lowest terms. If the rational function f ( x ) approaches either positive or negative infinity as x approaches a real number a from the right or the left, we call the line x = a a vertical asymptote of f ( x ). The rational function f ( x ) = p ( x ) q ( x ) has vertical asymptotes where q ( x ) = 0. If the degree of the polynomial in the numerator, p ( x ), is one more than the degree of the polynomial in the denominator, q ( x ), we can express f ( x ) as f ( x ) = a x + b + r ( x ) q ( x ) using the division algorithm. In such case, as x approaches positive or negative infinity, r ( x ) q ( x ) approaches 0. Hence, as | x | becomes large, r ( x ) approaches the line y = a x + b . The line y = a x + b is a slant asymptote or an oblique asymptote of the rational function r ( x ). The following are the procedures to be followed to graph the rational function f ( x ): Step 1: Factor the polynomial p ( x ) and q ( x ). Step 2: Find the vertical asymptotes (if any) by equating the denominator of f ( x ) to 0. Step 3: Find the horizontal asymptotes (if any). Step 4: Find the slant asymptote if the degree of p ( x ) is one more than the degree of q ( x ). Step 5: Find the x -intercept by solving the equation f ( x ) = 0. Step 6: find the y -intercept by calculating f (0). Step 7: Graph the intercepts and asymptotes of f ( x ). Plot some additional points and complete the graph of f ( x ). Given Information: The rational function r ( x ) = x 3 + x 2 x 2 − 4 . Calculation: Equate the denominator of r ( x ) to 0 and solve for x . ⇒ x 2 − 4 = 0 ⇒ x 2 = 4 ⇒ x = ± 2 Therefore, the vertical asymptote of r ( x ) is x = − 2 and x = 2. To analyze the sign of r ( x ) for x -values near the vertical asymptotes, we select the test values close to the left and right of − 2 and 2 check whether the sign of r ( x ) is positive or negative. Let x = − 2.1. Then r ( − 2.1 ) = ( − 2.1 ) 3 + ( − 2.1 ) 2 ( − 2.1 ) 2 − 4 . The sign of r ( x ) is ( − ) ( + ) which is positive. Let x = − 1.9. Then r ( − 1.9 ) = ( − 1.9 ) 3 + ( − 1.9 ) 2 ( − 1.9 ) 2 − 4 . The sign of r ( x ) is ( − ) ( − ) which is positive. Let x = 1.9. Then r ( 1.9 ) = ( 1.9 ) 3 + ( 1.9 ) 2 ( 1.9 ) 2 − 4 . The sign of r ( x ) is ( + ) ( − ) which is negative. Let x = 2

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ISBN: 8220100548512

Author: Watson

Publisher: CENGAGE L

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Chapter 3.6, Problem 75E

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The slant asymptote and the vertical asymptotes of the rational function r(x)=x3+x2x2−4 and sketch the graph of r(x).

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Chapter 3.6, Problem 74E

Chapter 3.6, Problem 76E

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The slant asymptote and the vertical asymptotes of the rational function r ( x ) = x 3 + x 2 x 2 − 4 and sketch the graph of r ( x ).

Solution Summary: The author describes the procedures to be followed to graph the rational function f ( x ).

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To find:

The slant asymptote and the vertical asymptotes of the rational function r(x)=x3+x2x2−4 and sketch the graph of r(x).

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Chapter 3 Solutions

The slant asymptote and the vertical asymptotes of the rational function r ( x ) = x 3 + x 2 x 2 − 4 and sketch the graph of r ( x ).

Solution Summary: The author describes the procedures to be followed to graph the rational function f ( x ).

Videos

To find: The slant asymptote and the vertical asymptotes of the rational function r(x)=x3+x2x2−4 and sketch the graph of r(x).

Want to see the full answer?

Chapter 3 Solutions

To find:

The slant asymptote and the vertical asymptotes of the rational function r(x)=x3+x2x2−4 and sketch the graph of r(x).