Question

algebra 2 362403 - 6 fy - 1
benchmark test 2
which graph(s) have a horizontal asymptote at y = -3? select all that apply.
a. $y = \frac{1}{x} - 3$
b. $y = \frac{-3x}{x - 3}$
c. $y = \frac{3}{x + 3}$
d. $y = \frac{3x}{x + 3}$
question 30

Explanation:

Step1: Recall horizontal asymptote rules

For rational functions:

1. If degree of numerator < degree of denominator: horizontal asymptote $y=0$
2. If degree of numerator = degree of denominator: horizontal asymptote $y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$
3. For transformed rational functions, adjust for vertical shifts.

Step2: Analyze Option A

Function: $y = \frac{1}{x} - 3$
Base function $\frac{1}{x}$ has asymptote $y=0$. Vertical shift down 3 gives:
$y = 0 - 3 = -3$

Step3: Analyze Option B

Function: $y = \frac{-3x}{x - 3}$
Degrees of numerator/denominator = 1/1.
Horizontal asymptote:
$y = \frac{-3}{1} = -3$

Step4: Analyze Option C

Function: $y = \frac{3}{x + 3}$
Degree of numerator < denominator.
Horizontal asymptote:
$y = 0$

Step5: Analyze Option D

Function: $y = \frac{3x}{x + 3}$
Degrees of numerator/denominator = 1/1.
Horizontal asymptote:
$y = \frac{3}{1} = 3$

Answer:

A. $y = \frac{1}{x} - 3$, B. $y = \frac{-3x}{x - 3}$