Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=\frac{x + 2}{x - 2} ).
find the ( y )-intercepts of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the ( y )-intercept(s) is/are at ( y = - 1 ).
(type an integer or a decimal. use a comma to separate answers as needed.)
b. there are no ( y )-intercepts.
find any horizontal asymptotes of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
a. the function has one horizontal asymptote, .
(type an equation.)
b. the function has two horizontal asymptotes. the top asymptote is and the bottom asymptote is
(type equations.)
c. there are no horizontal asymptotes

Explanation:

Step1: Find the y - intercept

The y - intercept is found by setting \(x = 0\) in the function \(y=f(x)=\frac{x + 2}{x-2}\).
Substitute \(x = 0\) into the function: \(y=\frac{0 + 2}{0-2}=\frac{2}{-2}=-1\).

Step2: Find the horizontal asymptote

For a rational function \(y=\frac{ax^{n}+...}{bx^{m}+...}\), when \(n=m\) (here \(n = m=1\) for \(y=\frac{x + 2}{x-2}\)), the horizontal asymptote is \(y=\frac{a}{b}\).
Here \(a = 1\) (coefficient of \(x\) in the numerator) and \(b = 1\) (coefficient of \(x\) in the denominator), so \(y=\frac{1}{1}=1\).

Answer:

• For the y - intercept: A. The y - intercept(s) is/are at \(y=-1\)
• For the horizontal asymptote: A. The function has one horizontal asymptote, \(y = 1\)