Question

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.
$f(x) = -1 + \frac{2}{x + 6}$
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

Vertical asymptote occurs where the denominator is zero (and numerator non - zero). For \(f(x)=- 1+\frac{2}{x + 6}=\frac{-x - 6+2}{x + 6}=\frac{-x - 4}{x + 6}\), set denominator \(x + 6=0\), so \(x=-6\).

Step2: Find Horizontal Asymptote

For rational functions, if the degrees of numerator and denominator are equal, the horizontal asymptote is the ratio of leading coefficients. Here, rewrite \(f(x)=\frac{-x - 4}{x + 6}\), degree of numerator and denominator is 1. Leading coefficient of numerator is \(-1\), denominator is \(1\), so horizontal asymptote \(y=\frac{-1}{1}=-1\).

Step3: Find x - intercept

Set \(f(x) = 0\), so \(0=-1+\frac{2}{x + 6}\). Add 1 to both sides: \(1=\frac{2}{x + 6}\). Cross - multiply: \(x + 6=2\), then \(x=2 - 6=-4\). So x - intercept is \((-4,0)\).

Step4: Find y - intercept

Set \(x = 0\), \(f(0)=-1+\frac{2}{0 + 6}=-1+\frac{1}{3}=-\frac{2}{3}\). So y - intercept is \((0,-\frac{2}{3})\).

Answer:

s:

• Vertical Asymptote: \(x=-6\)
• Horizontal Asymptote: \(y = - 1\)
• x - intercept: \((-4,0)\)
• y - intercept: \((0,-\frac{2}{3})\)