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

4th Edition

ISBN: 8220100548512

Author: Watson

Publisher: CENGAGE L

See similar textbooks

Videos

Question

Chapter 3.6, Problem 75E

To determine

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).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 3.6, Problem 74E

Chapter 3.6, Problem 76E

Students have asked these similar questions

x²-x+1 Find the asymptotes of the function x-1

VERTICAL ASYMPTOTES Determine the vertical asymptotes for the function. Set the denominator equal to zero to find the x-value for which the function is undefined. That will be the vertical asymptote given the numerator does not equal 0 also (Remember this is called removable discontinuity). Write a vertical asymptotes as a line in the form x= Example: Find the vertical asymptote of y=- 1 x-2 Since when x=2 the function is in the form 1/0 then the vertical line x=2 is a vertical asymptote of the function. 44. f(x)== / 4-x 47. f(x)=x²-16 x² x²-4 45. f(x)=- x-1 x²+x-2 48. f(x)=- 12 =0 x-2 46. f(x)= 2+x x² (1-x) Sx-+20 x²-16 49. f(x)=-

49-60. Evaluate each limit. 56. lim 0→0 1 2+sin 0 sin 60. lim I⇒e² 11123 2 ln²x − 5 ln x + 6 - In x - 2

Chapter 3 Solutions

EBK ALGEBRA AND TRIGONOMETRY

Ch. 3.1 - To put the quadratic function fx=ax2+bx+c in...Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - 5-8 Graphs of Quadratic Functions The graph of a...Ch. 3.1 - Prob. 6E Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Prob. 9E Ch. 3.1 - Prob. 10E

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

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

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).