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

For a rational function, vertical asymptote occurs where the denominator is zero (and numerator is non - zero). In the function \(f(x)=- 1+\frac{4}{x - 4}=\frac{-1\times(x - 4)+4}{x - 4}=\frac{-x + 4 + 4}{x - 4}=\frac{-x+8}{x - 4}\), the denominator \(x - 4 = 0\) when \(x = 4\). So the vertical asymptote is \(x = 4\).

Step2: Find Horizontal Asymptote

For a rational function of the form \(y=\frac{ax + b}{cx + d}\), if the degrees of the numerator and denominator are equal, the horizontal asymptote is \(y=\frac{a}{c}\). First, rewrite \(f(x)\) as \(f(x)=\frac{-x + 8}{x - 4}\). The degree of the numerator (highest power of \(x\)) is 1, and the degree of the denominator is also 1. So the horizontal asymptote is \(y=\frac{-1}{1}=-1\).

Step3: Find x - intercept

To find the x - intercept, set \(y = f(x)=0\). So \(0=-1+\frac{4}{x - 4}\). Add 1 to both sides: \(1=\frac{4}{x - 4}\). Cross - multiply: \(x - 4=4\). Then \(x=4 + 4=8\). So the x - intercept is at the point \((8,0)\).

Step4: Find y - intercept

To find the y - intercept, set \(x = 0\). Then \(f(0)=-1+\frac{4}{0 - 4}=-1-1=-2\). So the y - intercept is at the point \((0,-2)\).

Answer:

• Vertical Asymptote: \(x = 4\)
• Horizontal Asymptote: \(y=-1\)
• x - intercept: \((8,0)\)
• y - intercept: \((0,-2)\)

To plot:

• Draw a vertical dashed line at \(x = 4\) (vertical asymptote).
• Draw a horizontal dashed line at \(y=-1\) (horizontal asymptote).
• Plot the point \((8,0)\) (x - intercept) and \((0,-2)\) (y - intercept). Then, using these features, you can sketch the graph of the rational function.