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.
$f(x) = \frac{-2}{x + 3} - 3$
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). For \( f(x)=\frac{-2}{x + 3}-3 \), the denominator is \( x + 3 \). Set \( x+3 = 0 \), so \( x=-3 \).

Step2: Find Horizontal Asymptote

For rational functions, if the degree of the numerator and denominator are equal (or we can rewrite the function). First, rewrite \( f(x)=\frac{-2}{x + 3}-3=\frac{-2-3(x + 3)}{x + 3}=\frac{-2-3x-9}{x + 3}=\frac{-3x-11}{x + 3} \). The degrees of numerator and denominator are both 1. The horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of numerator is -3, denominator is 1, so horizontal asymptote is \( y=-3 \).

Step3: Find x - intercept

Set \( f(x) = 0 \): \( \frac{-2}{x + 3}-3=0 \). Add 3 to both sides: \( \frac{-2}{x + 3}=3 \). Cross - multiply: \( -2 = 3(x + 3) \). Expand: \( -2=3x + 9 \). Subtract 9: \( 3x=-11 \), so \( x=-\frac{11}{3}\approx - 3.67 \).

Step4: Find y - intercept

Set \( x = 0 \): \( f(0)=\frac{-2}{0 + 3}-3=\frac{-2}{3}-3=\frac{-2 - 9}{3}=-\frac{11}{3}\approx - 3.67 \).

Answer:

• Vertical Asymptote: \( x=-3 \)
• Horizontal Asymptote: \( y = - 3 \)
• x - intercept: \( x=-\frac{11}{3}\) (or approximately - 3.67)
• y - intercept: \( y=-\frac{11}{3}\) (or approximately - 3.67)