To determine the vertical asymptotes of the function [tex]\( f(a) = \frac{a-4}{4a^2 + 7a - 2} \)[/tex], we need to identify where the denominator is equal to zero because vertical asymptotes occur where the function becomes undefined due to division by zero.
1. Identify the denominator:
The denominator of the function is [tex]\( 4a^2 + 7a - 2 \)[/tex].
2. Set the denominator equal to zero to find the critical points:
[tex]\[
4a^2 + 7a - 2 = 0
\][/tex]
3. Solve the quadratic equation for [tex]\(a\)[/tex]:
We solve the quadratic equation using the quadratic formula:
[tex]\[
a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation, this means:
[tex]\[
a = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 4 \cdot (-2)}}{2 \cdot 4}
\][/tex]
Simplifying inside the square root gives:
[tex]\[
a = \frac{-7 \pm \sqrt{49 + 32}}{8}
\][/tex]
Simplifying further:
[tex]\[
a = \frac{-7 \pm \sqrt{81}}{8}
\][/tex]
Since [tex]\( \sqrt{81} = 9 \)[/tex], we get:
[tex]\[
a = \frac{-7 \pm 9}{8}
\][/tex]
4. Find the two possible values for [tex]\(a\)[/tex]:
[tex]\[
a = \frac{-7 + 9}{8} = \frac{2}{8} = \frac{1}{4}
\][/tex]
and
[tex]\[
a = \frac{-7 - 9}{8} = \frac{-16}{8} = -2
\][/tex]
Hence, the vertical asymptotes occur at [tex]\( a = \frac{1}{4} \)[/tex] and [tex]\( a = -2 \)[/tex].
5. Express the vertical asymptotes as equations:
[tex]\[
a = \frac{1}{4}, \quad a = -2
\][/tex]
Thus, the equations of the vertical asymptotes are:
[tex]\[
a = \frac{1}{4}, \quad a = -2
\][/tex]
