Question

1. write a function of the form $f(x) = \frac{a}{x - h} + k$ with a vertical asymptote at $x = -15$ and a horizontal asymptote at $y = -6$.

Explanation:

Step1: Identify h from vertical asymptote

For the function \( f(x)=\frac{a}{x - h}+k \), the vertical asymptote is \( x = h \). Given vertical asymptote \( x=-15 \), so \( h=-15 \).

Step2: Identify k from horizontal asymptote

The horizontal asymptote of \( f(x)=\frac{a}{x - h}+k \) is \( y = k \). Given horizontal asymptote \( y=-6 \), so \( k = - 6 \).

Step3: Choose a value for a (non - zero)

We can choose \( a = 1 \) (any non - zero real number will work). Then the function is \( f(x)=\frac{1}{x-(-15)}-6=\frac{1}{x + 15}-6 \).

Answer:

\( f(x)=\frac{1}{x + 15}-6 \) (a can be any non - zero real number, here we chose \( a = 1 \))