We are tasked with finding the equation of a parabola using quadratic regression, which passes through the points $(1, -12)$, $(2, -7)$, and $(5, 80)$.

We assume a quadratic equation of the form:

$y = ax^2 + bx + c$

Step 1: Set up the system of equations

By substituting the given points into the equation $y = ax^2 + bx + c$, we get three equations:

1. For $(1, -12)$: $-12 = a(1)^2 + b(1) + c \quad \Rightarrow \quad -12 = a + b + c \quad \text{(Equation 1)}$

2. For $(2, -7)$: $-7 = a(2)^2 + b(2) + c \quad \Rightarrow \quad -7 = 4a + 2b + c \quad \text{(Equation 2)}$

3. For $(5, 80)$: $80 = a(5)^2 + b(5) + c \quad \Rightarrow \quad 80 = 25a + 5b + c \quad \text{(Equation 3)}$

Step 2: Solve the system of equations

We now have the following system of equations: