Question

what are the equations of the horizontal and vertical asymptotes of $y = \frac{7}{x + 9} + 3$?
horizontal asymptote:
vertical asymptote:
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Explanation:

Step1: Find Horizontal Asymptote

For a rational function of the form \( y=\frac{a}{x - h}+k \), the horizontal asymptote is \( y = k \). Here, the function is \( y=\frac{7}{x + 9}+3 \), so \( k = 3 \). Thus, the horizontal asymptote is \( y = 3 \).

Step2: Find Vertical Asymptote

For a rational function \( y=\frac{a}{x - h}+k \), the vertical asymptote is found by setting the denominator equal to zero and solving for \( x \). The denominator here is \( x + 9 \). Setting \( x+9 = 0 \), we get \( x=-9 \). So the vertical asymptote is \( x=-9 \).

Answer:

horizontal asymptote: \( y = 3 \)
vertical asymptote: \( x=-9 \)