To find the value of [tex]\(a\)[/tex] for the quadratic function [tex]\(f(x)\)[/tex] with given zeros [tex]\(-8\)[/tex] and [tex]\(4\)[/tex] and a vertex at [tex]\((-2, 18)\)[/tex], we need to use the properties of the quadratic function in factored and vertex form.
The factored form of a quadratic function is:
[tex]\[ f(x) = a(x - x_1)(x - x_2) \][/tex]
Given the zeros [tex]\(-8\)[/tex] and [tex]\(4\)[/tex], the equation becomes:
[tex]\[ f(x) = a(x + 8)(x - 4) \][/tex]
We also know the vertex form of a quadratic function, which is:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex. Here, the vertex [tex]\((-2, 18)\)[/tex] gives us:
[tex]\[ f(x) = a(x + 2)^2 + 18 \][/tex]
Since the function [tex]\(f(x)\)[/tex] passes through the point [tex]\((-2, 18)\)[/tex], we equate both forms at [tex]\(x = -2\)[/tex]:
1. Substitute [tex]\(x = -2\)[/tex] into the factored form:
[tex]\[ f(-2) = a(-2 + 8)(-2 - 4) \][/tex]
Simplify within the parentheses:
[tex]\[ f(-2) = a(6)(-6) \][/tex]
[tex]\[ f(-2) = -36a \][/tex]
Given that [tex]\(f(-2) = 18\)[/tex]:
[tex]\[ 18 = -36a \][/tex]
To find [tex]\(a\)[/tex], solve the equation:
[tex]\[ a = -\frac{18}{36} \][/tex]
[tex]\[ a = -\frac{1}{2} \][/tex]
So, the correct value of [tex]\(a\)[/tex] is:
[tex]\[ \boxed{-\frac{1}{2}} \][/tex]
Hence, the correct answer is: B. [tex]\(-\frac{1}{2}\)[/tex]
