Question

algebra ii sem 2
1.9.3 quiz: finding vertical asymptotes
which of the following rational functions is graphed below?
a. $f(x)=\frac{1}{x}$
b. $f(x)=\frac{1}{(x-1)^2}$
c. $f(x)=\frac{1}{x^2-1}$

Explanation:

Step1: Identify vertical asymptote

The graph has a vertical asymptote at $x=1$. For a rational function, vertical asymptotes occur where the denominator equals 0 (and numerator is non-zero there).

Step2: Check denominator roots

• For A: Denominator $x=0$ gives asymptote $x=0$, does not match.
• For B: Denominator $(x-1)^2=0$ gives $x=1$, matches asymptote. Also, the graph is always positive (since square of denominator is positive, numerator 1 is positive), which matches the graph (all values above x-axis).
• For C: Denominator $x^2-1=(x-1)(x+1)=0$ gives asymptotes $x=1$ and $x=-1$, but the graph only has one vertical asymptote, so this does not match.

Answer:

B. $F(x)=\frac{1}{(x-1)^2}$