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.
$f(x) = -6 - \frac{3}{x + 6}$
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 the numerator is not zero there). For \( f(x)=-6 - \frac{3}{x + 6}=\frac{-6(x + 6)-3}{x + 6}=\frac{-6x-36 - 3}{x + 6}=\frac{-6x-39}{x + 6} \), set the denominator \( x + 6=0 \). Solving for \( x \), we get \( x=-6 \). So the vertical asymptote is \( x = - 6 \).

Step2: Find Horizontal Asymptote

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), if the degrees of \( N(x) \) and \( D(x) \) are equal, the horizontal asymptote is the ratio of the leading coefficients. Here, the degree of the numerator (after combining terms) is 1 and the degree of the denominator is 1. The leading coefficient of the numerator (for the linear term) is -6 and the leading coefficient of the denominator is 1. So the horizontal asymptote is \( y=-6 \).

Step3: Find x - intercept

To find the x - intercept, set \( f(x) = 0 \). So \( -6-\frac{3}{x + 6}=0 \). Add 6 to both sides: \( -\frac{3}{x + 6}=6 \). Multiply both sides by \( x + 6 \) (assuming \( x
eq - 6 \)): \( - 3=6(x + 6) \). Divide both sides by 6: \( -\frac{3}{6}=x + 6 \), which simplifies to \( -\frac{1}{2}=x + 6 \). Subtract 6 from both sides: \( x=-\frac{1}{2}-6=-\frac{1 + 12}{2}=-\frac{13}{2}=-6.5 \).

Step4: Find y - intercept

To find the y - intercept, set \( x = 0 \) in the function. \( f(0)=-6-\frac{3}{0 + 6}=-6-\frac{3}{6}=-6 - 0.5=-6.5 \).

Answer:

• Vertical Asymptote: \( x=-6 \)
• Horizontal Asymptote: \( y = - 6 \)
• x - intercept: \( x=-\frac{13}{2}=-6.5 \) (the point is \( (-6.5,0) \))
• y - intercept: \( y=-6.5 \) (the point is \( (0,-6.5) \))