Questions: Polynomial and Rational Functions Matching graphs with rational functions: Two vertical asymptotes (a) Which is the graph of f(x)=(2x^2+4x-6)/(x^2-4)? (choose one) (b) Which is the graph of g(x)=(2x-2)/(x^2-9)? (choose one)

Polynomial and Rational Functions
Matching graphs with rational functions: Two vertical asymptotes
(a) Which is the graph of f(x)=(2x^2+4x-6)/(x^2-4)?
(choose one)
(b) Which is the graph of g(x)=(2x-2)/(x^2-9)?
(choose one)

Transcript text: Polynomial and Rational Functions Matching graphs with rational functions: Two vertical asymptotes (a) Which is the graph of $f(x)=\frac{2 x^{2}+4 x-6}{x^{2}-4}$ ? (choose one) (b) Which is the graph of $g(x)=\frac{2 x-2}{x^{2}-9}$ ? (choose one)

Solution

Solution Steps

Step 1: Identify the vertical asymptotes of the function

For the function $ f(x) = \frac{2x^2 + 4x - 6}{x^2 - 4} $, the vertical asymptotes occur where the denominator is zero. Set the denominator equal to zero and solve for $ x $: \[ x^2 - 4 = 0 \] \[ x^2 = 4 \] \[ x = \pm 2 \]

Step 2: Identify the vertical asymptotes in the graphs

Examine the graphs to identify which ones have vertical asymptotes at $ x = 2 $ and $ x = -2 $.

Step 3: Match the function to the correct graph

Compare the identified vertical asymptotes with the graphs provided. The graph that has vertical asymptotes at $ x = 2 $ and $ x = -2 $ is the correct match for the function $ f(x) = \frac{2x^2 + 4x - 6}{x^2 - 4} $.