Question

find 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.
$f(x) = \frac{-1}{x + 4} - 1$
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 2

Explanation:

Step1: Find Vertical Asymptote

Vertical asymptote occurs where the denominator is zero (and numerator is not zero there). For \( f(x)=\frac{-1}{x + 4}-1=\frac{-1-(x + 4)}{x + 4}=\frac{-x-5}{x + 4} \), denominator \( x + 4 = 0\) when \( x=-4 \). Numerator at \( x=-4 \) is \( -(-4)-5=4 - 5=-1
eq0 \), so vertical asymptote is \( x=-4 \).

Step2: Find Horizontal Asymptote

For rational functions, if degrees of numerator and denominator are equal, horizontal asymptote is ratio of leading coefficients. Here, numerator \( -x - 5 \) (degree 1, leading coefficient -1) and denominator \( x + 4 \) (degree 1, leading coefficient 1). So horizontal asymptote is \( y=\frac{-1}{1}=-1 \).

Step3: Find x - intercept

Set \( f(x)=0 \), so \( \frac{-x - 5}{x + 4}=0 \). Numerator must be zero: \( -x - 5 = 0\Rightarrow x=-5 \). Check denominator at \( x=-5 \): \( -5 + 4=-1
eq0 \), so x - intercept is \( (-5,0) \).

Step4: Find y - intercept

Set \( x = 0 \), \( f(0)=\frac{-0 - 5}{0 + 4}=\frac{-5}{4}=-1.25 \), so y - intercept is \( (0,-\frac{5}{4}) \) or \( (0,-1.25) \).

Answer:

• Vertical Asymptote: \( x=-4 \)
• Horizontal Asymptote: \( y=-1 \)
• x - intercept: \( (-5,0) \)
• y - intercept: \( (0,-\frac{5}{4}) \) (or \( (0,-1.25) \))