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 poss
$f(x) = \frac{-6}{x + 5} - 6$
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

The vertical asymptote of a rational function occurs where the denominator is zero (and numerator is not zero). For \( f(x)=\frac{-6}{x + 5}-6 \), the denominator is \( x + 5 \). Set \( x+5 = 0 \), so \( x=-5 \).

Step2: Find Horizontal Asymptote

For rational functions, if the degree of numerator and denominator are equal (after simplifying), the horizontal asymptote is the ratio of leading coefficients. First, rewrite \( f(x) \) as a single fraction: \( f(x)=\frac{-6-6(x + 5)}{x + 5}=\frac{-6-6x-30}{x + 5}=\frac{-6x-36}{x + 5} \). The degree of numerator (\( -6x-36 \)) and denominator (\( x + 5 \)) is 1 (both are linear). The leading coefficient of numerator is -6, denominator is 1. So horizontal asymptote is \( y=\frac{-6}{1}=-6 \).

Step3: Find x - intercept

Set \( f(x) = 0 \): \( \frac{-6}{x + 5}-6=0 \). Add 6 to both sides: \( \frac{-6}{x + 5}=6 \). Cross - multiply: \( -6 = 6(x + 5) \). Divide both sides by 6: \( -1=x + 5 \). Subtract 5: \( x=-6 \). So x - intercept is at \( (-6,0) \).

Step4: Find y - intercept

Set \( x = 0 \) in \( f(x) \): \( f(0)=\frac{-6}{0 + 5}-6=\frac{-6}{5}-6=\frac{-6-30}{5}=\frac{-36}{5}=-7.2 \). So y - intercept is at \( (0,-7.2) \).

Answer:

• Vertical Asymptote: \( x=-5 \)
• Horizontal Asymptote: \( y = - 6 \)
• x - intercept: \( (-6,0) \)
• y - intercept: \( (0,-\frac{36}{5}) \) (or \( (0, - 7.2) \))