Question

determine the equation of the horizontal asymptote of the rational function. $f(x)=\frac{5x^{2}-2}{-5x^{2}+7}$
$y = 0$
$y = 1$
$y=-1$
$y = 5$
question #7
true or false. the equation of the slant asymptote of $y=\frac{x^{2}-2x + 1}{x + 4}$ will be $y=-x + 6$
true
false

Explanation:

Step1: Recall horizontal - asymptote rule

For a rational function $f(x)=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}$, when $n = m$, the horizontal asymptote is $y=\frac{a_n}{b_m}$.

Step2: Identify coefficients

In $f(x)=\frac{5x^2 - 2}{-5x^2+7}$, $n = m=2$, $a_n = 5$, $b_m=-5$.

Step3: Calculate horizontal asymptote

$y=\frac{5}{-5}=-1$.

Step4: Recall slant - asymptote rule

For a rational function $y=\frac{f(x)}{g(x)}$ where the degree of $f(x)$ is one more than the degree of $g(x)$, we use polynomial long - division. Divide $x^2-2x + 1$ by $x + 4$.

Step5: Perform polynomial long - division

$x^2-2x + 1=(x + 4)(x-6)+25$. So, $y=\frac{x^2-2x + 1}{x + 4}=x-6+\frac{25}{x + 4}$. The slant asymptote is $y=x - 6$, not $y=-x + 6$.

Answer:

1. C. $y=-1$
2. False