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

Explanation:

Step1: Find Vertical Asymptote

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

Step2: Find Horizontal Asymptote

For a rational function \(y=\frac{a_nx^n+\cdots}{b_mx^m+\cdots}\), if \(n=m\), horizontal asymptote is \(y=\frac{a_n}{b_m}\). Here, the degree of numerator (\(x + 4\)) and denominator (\(x - 2\)) is 1. So horizontal asymptote is \(y = 1\) (since the leading coefficients are both 1).

Step3: Find x - intercept

Set \(y = 0\), so \(0=1+\frac{6}{x - 2}\). Subtract 1: \(-1=\frac{6}{x - 2}\). Cross - multiply: \(-(x - 2)=6\). \(-x + 2=6\). \(-x=4\), so \(x=-4\). x - intercept is \((-4,0)\).

Step4: Find y - intercept

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

Answer:

• Vertical Asymptote: \(x = 2\)
• Horizontal Asymptote: \(y = 1\)
• x - intercept: \((-4,0)\)
• y - intercept: \((0,-2)\)

To plot: Drag the vertical asymptote to \(x = 2\), horizontal asymptote to \(y = 1\), x - intercept to \((-4,0)\) and y - intercept to \((0,-2)\) on the graph.