The problem asks us to find the equation of a quadratic function in factored form, given three points that lie on the graph of the function: $(1, 0)$, $(-2, 0)$, and $(-0.5, 1.125)$. The factored form is given as $y = a(x - p)(x - q)$.

1. Problem Description

The problem asks us to find the equation of a quadratic function in factored form, given three points that lie on the graph of the function: $(1, 0)$, $(-2, 0)$, and $(-0.5, 1.125)$. The factored form is given as $y = a(x - p)(x - q)$.

2. Solution Steps

Since the points $(1, 0)$ and $(-2, 0)$ are x-intercepts, we can say that $p = 1$ and $q = -2$ (or vice versa). Thus, the equation becomes $y = a(x - 1)(x - (-2))$, which simplifies to:

$y = a(x - 1)(x + 2)$

Now we need to find the value of $a$. We can use the point $(-0.5, 1.125)$ to solve for $a$. Substituting $x = -0.5$ and $y = 1.125$ into the equation, we get:

$1.125 = a(-0.5 - 1)(-0.5 + 2)$

$1.125 = a(-1.5)(1.5)$

$1.125 = a(-2.25)$

To solve for $a$, divide both sides by $-2.25$:

$a = \frac{1.125}{-2.25}$

$a = -\frac{1.125}{2.25}$

$a = -\frac{1125}{2250}$

$a = -\frac{1}{2}$

Now substitute $a = -\frac{1}{2}$ back into the factored form:

$y = -\frac{1}{2}(x - 1)(x + 2)$

3. Final Answer

y = -1/2(x - 1)(x + 2)