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.
$f(x) = \frac{-6}{x - 6} + 6$
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 \( f(x)=\frac{-6}{x - 6}+6 \), the denominator is \( x - 6 \). Set \( x - 6 = 0 \), so \( x = 6 \).

Step2: Find Horizontal Asymptote

For rational functions, if the degrees of numerator and denominator are equal, the horizontal asymptote is the ratio of leading coefficients. First, rewrite \( f(x) \) as a single fraction: \( f(x)=\frac{-6 + 6(x - 6)}{x - 6}=\frac{-6+6x - 36}{x - 6}=\frac{6x - 42}{x - 6} \). The degree of numerator and denominator is 1 (equal). The leading coefficient of numerator is 6, denominator is 1. So horizontal asymptote is \( y = \frac{6}{1}=6 \).

Step3: Find x - intercept

Set \( f(x)=0 \): \( \frac{-6}{x - 6}+6 = 0 \). Subtract 6: \( \frac{-6}{x - 6}=-6 \). Multiply both sides by \( x - 6 \): \( -6=-6(x - 6) \). Divide by -6: \( 1=x - 6 \), so \( x = 7 \). x - intercept is \( (7,0) \).

Step4: Find y - intercept

Set \( x = 0 \): \( f(0)=\frac{-6}{0 - 6}+6=\frac{-6}{-6}+6 = 1 + 6 = 7 \). y - intercept is \( (0,7) \).

Answer:

• Vertical Asymptote: \( x = 6 \)
• Horizontal Asymptote: \( y = 6 \)
• x - intercept: \( (7,0) \)
• y - intercept: \( (0,7) \)

To plot: Drag the vertical asymptote to \( x = 6 \), horizontal asymptote to \( y = 6 \), x - intercept to \( (7,0) \), y - intercept to \( (0,7) \) on the graph.