To determine the vertical asymptote(s) of the function [tex]\( a(x) = \frac{8}{x^2 - 4x + 4} \)[/tex], we need to find the values of [tex]\( x \)[/tex] that make the denominator zero, because a vertical asymptote occurs when the function approaches infinity as the denominator approaches zero.
1. Identify the denominator:
The denominator of the given function is [tex]\( x^2 - 4x + 4 \)[/tex].
2. Solve the equation:
Set the denominator equal to zero to find the values of [tex]\( x \)[/tex] that cause the function to have vertical asymptotes.
[tex]\[
x^2 - 4x + 4 = 0
\][/tex]
3. Factor the quadratic expression:
Notice that [tex]\( x^2 - 4x + 4 \)[/tex] can be factored as a perfect square trinomial.
[tex]\[
x^2 - 4x + 4 = (x - 2)^2
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Set the factored expression equal to zero.
[tex]\[
(x - 2)^2 = 0
\][/tex]
Solve for [tex]\( x \)[/tex] by taking the square root of both sides.
[tex]\[
x - 2 = 0
\][/tex]
[tex]\[
x = 2
\][/tex]
Therefore, the value [tex]\( x = 2 \)[/tex] makes the denominator zero, indicating that there is a vertical asymptote at [tex]\( x = 2 \)[/tex].
Thus, the vertical asymptote of the function [tex]\( a(x) = \frac{8}{x^2 - 4x + 4} \)[/tex] is:
[tex]\[
x = 2
\][/tex]
