Question

question
plot all intercepts and asymptotes of the following rational function. if you get a fraction or decimal then plot as close to the true location as possible.
$f(x) = \frac{-2}{x + 6} - 4$
plot rational function
vertical asymptote horizontal asymptote
x-intercept y-intercept
click on a feature then drag it into place.

Explanation:

Step1: Find Vertical Asymptote

The vertical asymptote of a rational function occurs where the denominator is zero (and numerator is non - zero). For \(f(x)=\frac{-2}{x + 6}-4=\frac{-2-4(x + 6)}{x + 6}=\frac{-2-4x-24}{x + 6}=\frac{-4x-26}{x + 6}\), set the denominator \(x + 6=0\).
\(x=-6\)

Step2: Find Horizontal Asymptote

For a rational function of the form \(y=\frac{ax + b}{cx + d}\) (after combining terms), if the degrees of the numerator and denominator are equal, the horizontal asymptote is \(y=\frac{a}{c}\). Here, after combining \(f(x)=\frac{-4x-26}{x + 6}\), the degree of numerator and denominator is 1. So, horizontal asymptote is \(y=\frac{-4}{1}=-4\)

Step3: Find x - intercept

Set \(y = f(x)=0\), so \(\frac{-2}{x + 6}-4 = 0\).
\(\frac{-2}{x + 6}=4\)
\(-2 = 4(x + 6)\)
\(-2=4x+24\)
\(4x=-2 - 24=-26\)
\(x=\frac{-26}{4}=-\frac{13}{2}=-6.5\)

Step4: Find y - intercept

Set \(x = 0\) in \(f(x)\), \(f(0)=\frac{-2}{0 + 6}-4=\frac{-2}{6}-4=\frac{-1}{3}-4=\frac{-1 - 12}{3}=\frac{-13}{3}\approx - 4.33\)

Answer:

Vertical Asymptote: \(x=-6\)
Horizontal Asymptote: \(y = - 4\)
x - intercept: \(x=-\frac{13}{2}\) (or \(x=-6.5\))
y - intercept: \(y =-\frac{13}{3}\) (or \(y\approx - 4.33\))