Question

question
plot all of the existing five features of the following rational function (some may not be needed). if you get a fraction or decimal then plot as close to the true location as possible.
$f(x) = \frac{3x - 12}{x^2 - 16}$
plot rational function
vertical asymptote horizontal asymptote
x-intercept y-intercept hole
click on a feature then drag it into place.
y

Explanation:

Step1: Simplify the function

Factor numerator and denominator: $f(x)=\frac{3(x-4)}{(x-4)(x+4)}$, cancel $(x-4)$ (x≠4) to get $f(x)=\frac{3}{x+4}$ (x≠4).

Step2: Find vertical asymptote

Denominator zero (after simplification): $x+4=0 \Rightarrow x=-4$.

Step3: Find horizontal asymptote

Degree of numerator < denominator: $y=0$.

Step4: Find x-intercept

Numerator zero (original, before canceling): $3x-12=0 \Rightarrow x=4$, but x=4 is a hole, so no x-intercept.

Step5: Find y-intercept

Set x=0: $f(0)=\frac{-12}{-16}=\frac{3}{4}=0.75$.

Step6: Find hole

Canceled factor: $x=4$, plug into simplified function: $f(4)=\frac{3}{4+4}=\frac{3}{8}=0.375$, so hole at (4, 0.375).

Answer:

Vertical Asymptote: $x=-4$
Horizontal Asymptote: $y=0$
y-Intercept: $(0, 0.75)$
Hole: $(4, 0.375)$
No x-intercept.