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:

Analyze the vertical asymptote

Using the Vertical Asymptotes Determination knowledge point

$$
LATEXBLOCK0
$$

Analyze the horizontal asymptote and x-intercept

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

$$
LATEXBLOCK1
$$

Determine the constant using the y-intercept

Using the Y-Intercept of Rational Function knowledge point

$$
LATEXBLOCK2
$$

Identify the correct graph

Using the Rational Function Evaluation knowledge point

$$
LATEXBLOCK3
$$

This matches the graph in option A, which lies below the x-axis to the left of \(x = 10\) and above the x-axis to the right of \(x = 10\).

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) curve below x-axis on left of x=10, above on right (Correct answer)
• (B) curve above x-axis on left of x=10, below on right
• (C) curve below x-axis on both sides of x=10
• (D) curve above x-axis on both sides of x=10