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.
$f(x) = 2 - \frac{1}{x + 2}$
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

Set denominator to 0: $x+2=0 \Rightarrow x=-2$

Step2: Find horizontal asymptote

Limit as $x\to\infty$: $\lim_{x\to\infty}(2-\frac{1}{x+2})=2$

Step3: Find x-intercept

Set $f(x)=0$: $2-\frac{1}{x+2}=0 \Rightarrow \frac{1}{x+2}=2 \Rightarrow x+2=\frac{1}{2} \Rightarrow x=-\frac{3}{2}$

Step4: Find y-intercept

Set $x=0$: $f(0)=2-\frac{1}{0+2}=2-\frac{1}{2}=\frac{3}{2}$

Answer:

Vertical Asymptote: $x=-2$
Horizontal Asymptote: $y=2$
x-intercept: $(-\frac{3}{2},0)$
y-intercept: $(0,\frac{3}{2})$

(Plot these features on the graph: vertical line at $x=-2$, horizontal line at $y=2$, point at $(-1.5,0)$, point at $(0,1.5)$)