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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:

\[
\begin{align*}
a + b + c &= -12 \quad \text{(Equation 1)} \\
4a + 2b + c &= -7 \quad \text{(Equation 2)} \\
25a + 5b + c &= 80 \quad \text{(Equation 3)}
\end{align*}
\]

I will solve this system to find the values of \(a\), \(b\), and \(c\).The solution to the system of equations is:

\[
a = 6, \quad b = -13, \quad c = -5
\]

Thus, the equation of the parabola that passes through the points \((1, -12)\), \((2, -7)\), and \((5, 80)\) is:

\[
y = 6x^2 - 13x - 5
\]

Would you like further details or explanations?

Here are 5 related questions you might find helpful:
1. How can I verify the quadratic equation using the given points?
2. What is the geometric interpretation of the coefficients \(a\), \(b\), and \(c\)?
3. How do you solve a system of equations involving quadratic functions manually?
4. How can I apply this method to solve other quadratic regression problems?
5. What if the points were not given exactly; how would quadratic regression handle errors?

**Tip**: When solving systems of equations, substitution or elimination methods are effective strategies depending on the complexity of the equations.

Use quadratic regression to find the equation for the parabola going through the points (1, -12), (2, -7), and (5, 80).

Learn how to use quadratic regression to find the equation of a parabola that passes through the points (1, -12), (2, -7), and (5, 80). Understand key concepts such as quadratic equations and solving systems of equations. Suitable for high school students.

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