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 lo
$f(x) = \frac{-2}{x - 6} - 2$
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 \( f(x)=\frac{g(x)}{h(x)} \) occurs where \( h(x) = 0 \) (and \( g(x)
eq0 \) at that point). For \( f(x)=\frac{-2}{x - 6}-2 \), we first simplify it. Combine the terms: \( f(x)=\frac{-2-2(x - 6)}{x - 6}=\frac{-2-2x + 12}{x - 6}=\frac{-2x+10}{x - 6} \). The denominator \( x - 6=0 \) when \( x = 6 \). So vertical asymptote is \( x = 6 \).

Step2: Find Horizontal Asymptote

For a rational function \( \frac{ax^n+...}{bx^m+...} \), if \( n=m \), horizontal asymptote is \( y=\frac{a}{b} \). Here, numerator degree \( n = 1 \), denominator degree \( m = 1 \), \( a=-2 \), \( b = 1 \), so horizontal asymptote is \( y=-2 \).

Step3: Find x - intercept

Set \( f(x)=0 \), so \( \frac{-2x + 10}{x - 6}=0 \). Numerator \( -2x+10 = 0\Rightarrow -2x=-10\Rightarrow x = 5 \).

Step4: Find y - intercept

Set \( x = 0 \) in \( f(x)=\frac{-2}{0 - 6}-2=\frac{-2}{-6}-2=\frac{1}{3}-2=\frac{1 - 6}{3}=-\frac{5}{3}\approx - 1.67 \).

Answer:

• Vertical Asymptote: \( x = 6 \)
• Horizontal Asymptote: \( y=-2 \)
• x - intercept: \( x = 5 \) (point \( (5,0) \))
• y - intercept: \( y=-\frac{5}{3} \) (point \( (0,-\frac{5}{3}) \))