Question

if f has a vertical asymptote given by x = 10, a horizontal asymptote y = 0, y-intercept at -2, and no x-intercept, then write the equation of a rational function f(x) = \frac{p(x)}{q(x)} having the indicated properties, in which the degrees of p and q are as small as possible. graph the function using a graphing utility to verify that it has the required properties.

write the equation of the rational function. choose the correct answer below.
a. f(x) = \frac{20}{x + 10}
b. f(x) = \frac{20}{x - 10}
c. f(x) = \frac{10}{x + 10}
d. f(x) = \frac{10}{x - 10}

use a graphing utility to graph the function obtained in the previous step. choose the correct graph below. each graph is shown in a viewing rectangle -20,20,4 by -20,20,4

Explanation:

Determine the denominator from the vertical asymptote

Using the Vertical Asymptotes Determination knowledge point

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LATEXBLOCK0
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Determine the numerator from the horizontal asymptote and intercepts

Using the Horizontal Asymptotes and X-Intercept of Rational Function knowledge points

$$
LATEXBLOCK1
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Solve for the constant using the y-intercept

Using the Y-Intercept of Rational Function knowledge point

$$
LATEXBLOCK2
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Identify the correct graph

Using the Vertical Asymptotes Determination and Y-Intercept of Rational Function knowledge points

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LATEXBLOCK3
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Comparing the options, Graph B shows the vertical asymptote at \(x = 10\), the y-intercept at \((0, -2)\), and the branches in the correct quadrants.

Answer:

Question 1

• (A) \(f(x) = \frac{20}{x + 10}\)
• (B) \(f(x) = \frac{20}{x - 10}\) (Correct answer)
• (C) \(f(x) = \frac{10}{x + 10}\)
• (D) \(f(x) = \frac{10}{x - 10}\)

Question 2

• (A) branch in upper-left and lower-right, asymptote at x = -10
• (B) branch in upper-right and lower-left, asymptote at x = 10 (Correct answer)
• (C) branch in upper-left and lower-right, asymptote at x = 10
• (D) branch in upper-right and lower-left, asymptote at x = -10