To determine the equation of the quadratic function [tex]\( f(x) \)[/tex] given its roots and a point on the function, let's follow a systematic approach:
1. Identify the roots of the quadratic function:
The roots of the quadratic function are given as [tex]\(-4\)[/tex] and [tex]\(2\)[/tex]. This means the quadratic function can be expressed in factored form as:
[tex]\[
f(x) = a(x - 2)(x + 4)
\][/tex]
where [tex]\( a \)[/tex] is a constant that we need to determine.
2. Use the given point to find [tex]\( a \)[/tex]:
The function passes through the point [tex]\((1, -5)\)[/tex]. This implies [tex]\( f(1) = -5 \)[/tex]. Substitute [tex]\( x = 1 \)[/tex] and [tex]\( f(x) = -5 \)[/tex] into the equation:
[tex]\[
-5 = a(1 - 2)(1 + 4)
\][/tex]
Simplify the right-hand side:
[tex]\[
-5 = a(-1)(5)
\][/tex]
[tex]\[
-5 = -5a
\][/tex]
Solve for [tex]\( a \)[/tex]:
[tex]\[
a = 1
\][/tex]
3. Write the complete equation of the function:
Now that we have determined [tex]\( a = 1 \)[/tex], we can write the final quadratic function as:
[tex]\[
f(x) = (x - 2)(x + 4)
\][/tex]
4. Verify the options:
Given the equation we derived, we need to check the provided options for correctness:
[tex]\[
f(x) = (x - 2)(x + 4)
\][/tex]
This matches the first option.
Therefore, the equation of the quadratic function [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = (x - 2)(x + 4)
\][/tex]
So, the correct option is:
[tex]\[
\boxed{1}
\][/tex]
