We are given a quadratic function $f(x)$ in the factored form $f(x) = a(x-p)(x-q)$. We are given the x-intercepts as $(-3,0)$ and $(1,0)$, and a point on the graph of the function as $(4, 21)$. We need to find the equation of the quadratic function.

1. Problem Description

We are given a quadratic function $f(x)$ in the factored form $f(x) = a(x-p)(x-q)$. We are given the x-intercepts as $(-3,0)$ and $(1,0)$, and a point on the graph of the function as $(4, 21)$. We need to find the equation of the quadratic function.

2. Solution Steps

The x-intercepts give us the values of $p$ and $q$. Since the x-intercepts are $-3$ and $1$, we have $p = -3$ and $q = 1$. Substituting these values into the factored form, we get:

$f(x) = a(x - (-3))(x - 1)$

$f(x) = a(x + 3)(x - 1)$

Now we need to find the value of $a$. We are given that the function passes through the point $(4, 21)$. This means that $f(4) = 21$. Substituting $x = 4$ into the equation, we have:

$21 = a(4 + 3)(4 - 1)$

$21 = a(7)(3)$

$21 = 21a$

Dividing both sides by 21, we get:

$a = 1$

Now that we have the value of $a$, we can write the equation of the quadratic function:

$f(x) = 1(x + 3)(x - 1)$

$f(x) = (x + 3)(x - 1)$

3. Final Answer

$f(x) = (x+3)(x-1)$