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.
$f(x) = \frac{-3}{x + 1} + 5$
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 \( f(x)=\frac{-3}{x + 1}+5\), the denominator is \(x + 1\). Set \(x+1 = 0\), so \(x=-1\).

Step2: Find Horizontal Asymptote

For rational functions of the form \(y=\frac{a}{x - h}+k\), the horizontal asymptote is \(y = k\). Here, \(k = 5\), so horizontal asymptote is \(y = 5\).

Step3: Find x - intercept

Set \(f(x)=0\):

$$
LATEXBLOCK0
$$

Step4: Find y - intercept

Set \(x = 0\):

$$ f(0)=\frac{-3}{0 + 1}+5=-3 + 5 = 2 $$

Answer:

• Vertical Asymptote: \(x=-1\)
• Horizontal Asymptote: \(y = 5\)
• x - intercept: \(x=-\frac{2}{5}\) (or \(x = - 0.4\))
• y - intercept: \(y = 2\)