Question

question 5 of 8 (1 point) | question attempt: 1 of unlimited (a) which is the graph of f(x)=\frac{2x - 2}{x^{2}-3x - 4}? select

Explanation:

Step1: Factor the denominator

Factor $x^{2}-3x - 4=(x - 4)(x+1)$. So $f(x)=\frac{2x - 2}{(x - 4)(x + 1)}$. The vertical - asymptotes occur where the denominator is zero, i.e., $x=4$ and $x=-1$.

Step2: Find the horizontal asymptote

Since the degree of the numerator ($n = 1$) is less than the degree of the denominator ($m = 2$), the horizontal asymptote is $y = 0$.

Step3: Find the $x$ - intercept

Set the numerator equal to zero: $2x-2=0$, so $x = 1$.

Step4: Analyze the sign of the function

We can use test points in the intervals $(-\infty,-1)$, $(-1,1)$, $(1,4)$ and $(4,\infty)$. For example, if $x=-2$, $f(-2)=\frac{2(-2)-2}{(-2 - 4)(-2+1)}=\frac{-4 - 2}{(-6)(-1)}=-1<0$. If $x = 0$, $f(0)=\frac{-2}{(-4)(1)}=\frac{1}{2}>0$. If $x=2$, $f(2)=\frac{2(2)-2}{(2 - 4)(2 + 1)}=\frac{2}{(-2)(3)}=-\frac{1}{3}<0$. If $x = 5$, $f(5)=\frac{2(5)-2}{(5 - 4)(5 + 1)}=\frac{8}{6}=\frac{4}{3}>0$.

Based on the vertical asymptotes $x=-1,x = 4$, horizontal asymptote $y = 0$, $x$-intercept at $x = 1$ and sign - analysis, we can identify the graph.

Answer:

Graph C