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.
$f(x) = \frac{-2}{x + 4} - 1$
plot rational function
vertical asymptote horizontal asymptote
x intercept y intercept
click on a feature then drag it into place.

Explanation:

Step1: Simplify the function

First, we simplify \( f(x)=\frac{-2}{x + 4}-1\) to a single fraction. We have \( f(x)=\frac{-2-(x + 4)}{x + 4}=\frac{-2 - x-4}{x + 4}=\frac{-x - 6}{x + 4}\)

Step2: Find Vertical Asymptote

The vertical asymptote of a rational function \(y = \frac{N(x)}{D(x)}\) occurs where \(D(x)=0\) (and \(N(x)
eq0\) at that point). For \(f(x)=\frac{-x - 6}{x + 4}\), set the denominator \(x + 4=0\), so \(x=-4\) is the vertical asymptote.

Step3: Find Horizontal Asymptote

For a rational function \(y=\frac{ax + b}{cx + d}\) (where the degrees of numerator and denominator are equal), the horizontal asymptote is \(y=\frac{a}{c}\). Here, the numerator is \(-x-6\) (degree 1, leading coefficient \(- 1\)) and the denominator is \(x + 4\) (degree 1, leading coefficient \(1\)). So the horizontal asymptote is \(y=\frac{-1}{1}=-1\)

Step4: Find x - intercept

To find the x - intercept, set \(y = f(x)=0\). So \(\frac{-x - 6}{x + 4}=0\). A fraction is zero when the numerator is zero (and denominator is not zero). Set \(-x-6 = 0\), then \(x=-6\). We check the denominator at \(x = - 6\): \(x+4=-6 + 4=-2
eq0\). So the x - intercept is at \(x=-6\) (the point \((-6,0)\))

Step5: Find y - intercept

To find the y - intercept, set \(x = 0\) in the function \(f(x)=\frac{-0 - 6}{0 + 4}=\frac{-6}{4}=-\frac{3}{2}=-1.5\). So the y - intercept is at the point \((0,-\frac{3}{2})\)

Answer:

• Vertical Asymptote: \(x=-4\)
• Horizontal Asymptote: \(y = - 1\)
• x - intercept: \((-6,0)\)
• y - intercept: \((0,-\frac{3}{2})\)

To plot:

• Draw the vertical line \(x=-4\) (vertical asymptote)
• Draw the horizontal line \(y=-1\) (horizontal asymptote)
• Plot the point \((-6,0)\) (x - intercept)
• Plot the point \((0,-\frac{3}{2})\) (y - intercept)