Alright, let's find the quadratic function \( f(x) \) given its roots and a point that lies on it.
### Step-by-Step Solution
1. Identify the form of the quadratic function:
Since the function has given roots \( -4 \) and \( 2 \), we can express it in the form:
[tex]\[
f(x) = a(x + 4)(x - 2)
\][/tex]
where \( a \) is a constant multiplier that we need to determine.
2. Substitute the given point \( (1, -5) \) into the equation:
This point means that when \( x = 1 \), \( f(x) = -5 \).
So, substitute \( x = 1 \) and \( f(x) = -5 \) into the quadratic equation:
[tex]\[
-5 = a(1 + 4)(1 - 2)
\][/tex]
3. Simplify the equation to solve for \( a \):
[tex]\[
-5 = a(5)(-1)
\][/tex]
[tex]\[
-5 = -5a
\][/tex]
Now, divide both sides by \(-5\) to isolate \( a \):
[tex]\[
a = 1
\][/tex]
4. Substitute \( a \) back into the quadratic equation:
Now that we've determined \( a \) is 1, we can substitute it back into the equation:
[tex]\[
f(x) = 1(x + 4)(x - 2)
\][/tex]
Which simplifies to:
[tex]\[
f(x) = (x + 4)(x - 2)
\][/tex]
5. Check the options provided:
Given the question's options, the correct one that matches our derived equation is:
[tex]\[
f(x) = (x - 2)(x + 4)
\][/tex]
Therefore, the equation of the quadratic function is:
[tex]\[
f(x) = (x - 2)(x + 4)
\][/tex]
