To determine the vertical asymptotes of the function [tex]\( f(x) = \frac{2x - 4}{x^2 - 4} \)[/tex], we follow these steps:
1. Identify the Denominator:
The denominator of the function is [tex]\( x^2 - 4 \)[/tex]. Vertical asymptotes occur where this denominator equals zero because division by zero is undefined.
2. Set the Denominator to Zero:
To find where the denominator equals zero, we solve the equation:
[tex]\[
x^2 - 4 = 0
\][/tex]
3. Factor the Denominator:
The expression [tex]\( x^2 - 4 \)[/tex] is a difference of squares and can be factored as follows:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
4. Solve for x:
Now, we solve each factor for zero:
[tex]\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\][/tex]
[tex]\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\][/tex]
5. Identify Vertical Asymptotes:
The values [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex] make the denominator zero, which means the function has vertical asymptotes at these points.
Thus, the function [tex]\( f(x) = \frac{2x - 4}{x^2 - 4} \)[/tex] has two vertical asymptotes.
Therefore, the correct answer is:
C. The graph has two vertical asymptotes.
