Question

bell ringer 1/21/26
find the equation of the slant asymptote of the rational function given below.
$f(x) = \frac{x^2 - 9x + 16}{-8 + x}$

Explanation:

Step1: Rewrite denominator

First, reorder the denominator terms to standard form:
$-8 + x = x - 8$
So the function becomes $f(x)=\frac{x^2 - 9x + 16}{x - 8}$

Step2: Polynomial long division

Divide $x^2 - 9x + 16$ by $x - 8$:

1. Divide $x^2$ by $x$: $x$
2. Multiply $x-8$ by $x$: $x^2 - 8x$
3. Subtract from dividend: $(x^2 - 9x + 16)-(x^2 - 8x)= -x + 16$
4. Divide $-x$ by $x$: $-1$
5. Multiply $x-8$ by $-1$: $-x + 8$
6. Subtract: $(-x + 16)-(-x + 8)=8$

So $f(x)=x - 1 + \frac{8}{x - 8}$

Step3: Identify slant asymptote

As $x\to\pm\infty$, $\frac{8}{x - 8}\to0$. The slant asymptote is the polynomial part.

Answer:

$y = x - 1$