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.
$f(x) = -5 + \frac{6}{x - 4}$
plot rational function
vertical asymptote horizontal asymptote
x - intercept y - intercept
click on a feature then drag it into place.
answer attempt 1 out of 3
you must answer all questions above in order to submit.

Explanation:

Step1: Find Vertical Asymptote

A vertical asymptote occurs where the denominator is zero (and numerator non - zero). For \(f(x)=- 5+\frac{6}{x - 4}=\frac{-5(x - 4)+6}{x - 4}=\frac{-5x + 20 + 6}{x - 4}=\frac{-5x+26}{x - 4}\), the denominator \(x - 4 = 0\) when \(x = 4\). So the vertical asymptote is \(x = 4\).

Step2: Find Horizontal Asymptote

For a rational function \(y=\frac{ax^{n}+...}{bx^{m}+...}\), if \(n=m\), the horizontal asymptote is \(y=\frac{a}{b}\). Here, rewrite \(f(x)\) as \(f(x)=\frac{-5x + 26}{x - 4}\), the degree of numerator (\(n = 1\)) and denominator (\(m = 1\)). So \(y=\frac{-5}{1}=-5\) is the horizontal asymptote.

Step3: Find x - intercept

Set \(y = 0\), so \(0=-5+\frac{6}{x - 4}\). Add 5 to both sides: \(5=\frac{6}{x - 4}\). Cross - multiply: \(5(x - 4)=6\), \(5x-20 = 6\), \(5x=26\), \(x=\frac{26}{5}=5.2\).

Step4: Find y - intercept

Set \(x = 0\), then \(f(0)=-5+\frac{6}{0 - 4}=-5-\frac{6}{4}=-5 - 1.5=-6.5\).

Answer:

• Vertical Asymptote: \(x = 4\)
• Horizontal Asymptote: \(y=-5\)
• x - intercept: \(x=\frac{26}{5}=5.2\) (point \((5.2,0)\))
• y - intercept: \(y = - 6.5\) (point \((0,-6.5)\))