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
$f(x) = -5 + \frac{1}{x - 1}$
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

A vertical asymptote occurs where the denominator is zero (and numerator non - zero). For \(f(x)=-5+\frac{1}{x - 1}=\frac{-5(x - 1)+1}{x - 1}=\frac{-5x+5 + 1}{x - 1}=\frac{-5x + 6}{x - 1}\), the denominator \(x-1 = 0\) when \(x = 1\). So the vertical asymptote is \(x=1\).

Step2: Find Horizontal Asymptote

For a rational function \(y=\frac{a_nx^n+\cdots}{b_mx^m+\cdots}\), if \(n=m\), the horizontal asymptote is \(y=\frac{a_n}{b_m}\); if \(n\lt m\), \(y = 0\); if \(n\gt m\), no horizontal asymptote. Here, we can rewrite \(f(x)=-5+\frac{1}{x - 1}\). As \(x\to\pm\infty\), \(\frac{1}{x - 1}\to0\), so \(f(x)\to - 5\). Thus, the horizontal asymptote is \(y=-5\).

Step3: Find x - intercept

Set \(y = f(x)=0\), so \(0=-5+\frac{1}{x - 1}\). Then \(\frac{1}{x - 1}=5\), cross - multiply to get \(1 = 5(x - 1)\), \(1=5x-5\), \(5x=6\), \(x=\frac{6}{5}=1.2\).

Step4: Find y - intercept

Set \(x = 0\) in \(f(x)\), \(f(0)=-5+\frac{1}{0 - 1}=-5-1=-6\).

Answer:

• Vertical Asymptote: \(x = 1\)
• Horizontal Asymptote: \(y=-5\)
• x - intercept: \(x=\frac{6}{5}\) (or \(x = 1.2\))
• y - intercept: \(y=-6\)

To plot:

• Drag the "Vertical Asymptote" to \(x = 1\) on the x - axis.
• Drag the "Horizontal Asymptote" to \(y=-5\) on the y - axis.
• Drag the "x - intercept" to \((1.2,0)\) on the x - axis.
• Drag the "y - intercept" to \((0,-6)\) on the y - axis.