Question

Sketch the graph of each rational function showing all the key features. Verify your graph by graphing the function on
the graphing calculator.
2. f(x) = (3x − 6)(x − 4) / x(x − 4)

Answer 1

Answer:

The x-intercept is 2.

The function has no y-intercept.

The vertical asymptote of the function is x=0.

The horizontal asymptote of the function is x=0.

The function has hole at x=4.

Step-by-step explanation:

The given function is

$f\left(x\right)=\dfrac{(3x-6)(x-4)}{x(x-4)}$

Cancel out common factors.

$f\left(x\right)=\dfrac{3x-6}{x}$

(i) x-intercept.

Substitute f(x)=0, in the given function.

$0=\dfrac{3x-6}{x}$

$0=3x-6$

$-3x=-6$

$x=2$

The x-intercept is 2.

(i) y-intercept.

Substitute x=0, in the given function.

$f\left(x\right)=\dfrac{3(0)-6}{(0)}=\infty$

The function has no y-intercept.

(iii) Vertical asymptote.

Equate the denominator equal to 0.

$x=0$

Therefore, the vertical asymptote of the function is x=0.

(iv) Horizontal asymptote.

If degree of numerator and denominator are same, then horizontal asymptote is

$y=\frac{\text{Leading coefficient of numerator}}{\text{Leading coefficient of denominator}}$

$y=\frac{3}{1}$

$y=3$

Therefore, the horizontal asymptote of the function is x=0.

(v) End behavior

$f(x)\rightarrow 3\text{ as }x\rightarrow -\infty$

$f(x)\rightarrow \infty\text{ as }\rightarrow 0^{-}$

$f(x)\rightarrow -\infty\text{ as }\rightarrow 0^{+}$

$f(x)\rightarrow 3\text{ as }\rightarrow \infty$

(vi) holes

Equate the cancel factors equal to 0, to find the holes.

$x-4=0$

$x=4$

The function has hole at x=4.