To determine the equation of the quadratic function [tex]\( f(x) \)[/tex] with roots [tex]\(-2\)[/tex] and [tex]\(3\)[/tex] and which passes through the point [tex]\((2, -4)\)[/tex], follow these steps:
1. General Form: Given that the function has roots [tex]\(-2\)[/tex] and [tex]\(3\)[/tex], we can write it in its factorized form:
[tex]\[
f(x) = a(x + 2)(x - 3)
\][/tex]
where [tex]\( a \)[/tex] is a constant that we need to determine.
2. Substitution of the Point:
We know that the point [tex]\((2, -4)\)[/tex] lies on [tex]\( f(x) \)[/tex]. This means when [tex]\( x = 2 \)[/tex], [tex]\( f(x) = -4 \)[/tex].
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( f(x) = -4 \)[/tex] into the equation:
[tex]\[
-4 = a(2 + 2)(2 - 3)
\][/tex]
3. Solve for [tex]\( a \)[/tex]:
Simplify the expression:
[tex]\[
-4 = a(4)(-1)
\][/tex]
[tex]\[
-4 = -4a
\][/tex]
Divide both sides by [tex]\(-4\)[/tex]:
[tex]\[
1 = a
\][/tex]
4. Formulate the Final Equation:
Now that we know [tex]\( a = 1 \)[/tex], we substitute [tex]\( a \)[/tex] back into the factorized form:
[tex]\[
f(x) = (x + 2)(x - 3)
\][/tex]
5. Conclusion:
Comparing this result with the given options, we find that the correct function is:
[tex]\[
f(x) = (x + 2)(x - 3)
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{(x+2)(x-3)}
\][/tex]
