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.
$f(x) = -1 + \frac{6}{x - 4}$
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

Vertical asymptote occurs where the denominator is zero (and numerator non - zero). For \(f(x)=- 1+\frac{6}{x - 4}=\frac{-x + 4+6}{x - 4}=\frac{-x + 10}{x - 4}\), the denominator \(x - 4=0\) when \(x = 4\). So vertical asymptote is \(x = 4\).

Step2: Find Horizontal Asymptote

For a rational function \(y=\frac{ax^n+...}{bx^m+...}\), if \(n=m\), horizontal asymptote is \(y=\frac{a}{b}\). Here, rewrite \(f(x)=\frac{-x + 10}{x - 4}\), the degree of numerator and denominator is 1. So horizontal asymptote is \(y=\frac{-1}{1}=-1\).

Step3: Find x - intercept

Set \(f(x) = 0\), so \(0=-1+\frac{6}{x - 4}\). Add 1 to both sides: \(1=\frac{6}{x - 4}\). Cross - multiply: \(x - 4=6\), then \(x=6 + 4=10\). So x - intercept is \((10,0)\).

Step4: Find y - intercept

Set \(x = 0\) in \(f(x)\), \(f(0)=-1+\frac{6}{0 - 4}=-1-\frac{6}{4}=-1-\frac{3}{2}=-\frac{5}{2}=-2.5\). So y - intercept is \((0,-2.5)\).

Answer:

• Vertical Asymptote: \(x = 4\)
• Horizontal Asymptote: \(y=-1\)
• x - intercept: \((10,0)\)
• y - intercept: \((0,-\frac{5}{2})\) (or \((0, - 2.5)\))