Question

epts and asymptotes of the following rational function. if you get a fractio
$f(x) = \frac{-6}{x + 5} - 6$
plot rational function
vertical asymptote horizontal asymptote
x-intercept y-intercept
click on a feature then drag it into place.
graph with x and y axes from -10 to 10

Explanation:

Step1: Find Vertical Asymptote

For a rational function, vertical asymptote occurs where the denominator is zero (and numerator is non - zero). The function is \(f(x)=\frac{-6}{x + 5}-6=\frac{-6-6(x + 5)}{x + 5}=\frac{-6-6x-30}{x + 5}=\frac{-6x-36}{x + 5}\). The denominator \(x + 5=0\) when \(x=-5\). So vertical asymptote is \(x=-5\).

Step2: Find Horizontal Asymptote

For a rational function \(y=\frac{ax + b}{cx + d}\) (after simplifying), if the degrees of numerator and denominator are equal, the horizontal asymptote is \(y=\frac{a}{c}\). Here, after simplifying \(f(x)=\frac{-6x-36}{x + 5}\), the degree of numerator (1) and denominator (1) are equal. So horizontal asymptote is \(y=\frac{-6}{1}=-6\).

Step3: Find x - Intercept

Set \(y = 0\), so \(\frac{-6x-36}{x + 5}=0\). The numerator \(-6x-36 = 0\) gives \(-6x=36\), so \(x=-6\). So x - intercept is \((-6,0)\).

Step4: Find y - Intercept

Set \(x = 0\), \(f(0)=\frac{-6}{0 + 5}-6=\frac{-6}{5}-6=\frac{-6-30}{5}=\frac{-36}{5}=-7.2\). So y - intercept is \((0,-\frac{36}{5})\) or \((0, - 7.2)\).

Answer:

Vertical Asymptote: \(x=-5\), Horizontal Asymptote: \(y = - 6\), x - Intercept: \((-6,0)\), y - Intercept: \((0,-\frac{36}{5})\)