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To solve this problem, we need to find the equation of a quadratic function \( y = ax^2 + bx + c \) that passes through the given points (-13, -126), (-7, 24), and (3, -46).

### Step 1: Set up a system of equations
For each point, substitute the values of \(x\) and \(y\) into the quadratic equation \( y = ax^2 + bx + c \).

1. For the point (-13, -126):
   \[
   -126 = a(-13)^2 + b(-13) + c
   \]
   \[
   -126 = 169a - 13b + c
   \]
   (Equation 1)

2. For the point (-7, 24):
   \[
   24 = a(-7)^2 + b(-7) + c
   \]
   \[
   24 = 49a - 7b + c
   \]
   (Equation 2)

3. For the point (3, -46):
   \[
   -46 = a(3)^2 + b(3) + c
   \]
   \[
   -46 = 9a + 3b + c
   \]
   (Equation 3)

### Step 2: Solve the system of equations

Now, we have the following system of equations:
1. \( 169a - 13b + c = -126 \)
2. \( 49a - 7b + c = 24 \)
3. \( 9a + 3b + c = -46 \)

We can solve this system using substitution, elimination, or matrix methods to find the values of \(a\), \(b\), and \(c\).

I'll calculate the values now.

### Step 3: Solution

Let me compute the exact values of \(a\), \(b\), and \(c\).The solution to the system of equations gives us the values of \(a\), \(b\), and \(c\):

\[
a = -2, \quad b = -15, \quad c = 17
\]

Thus, the quadratic equation for the parabola passing through the points \((-13, -126)\), \((-7, 24)\), and \((3, -46)\) is:

\[
y = -2x^2 - 15x + 17
\]

Would you like further details or have any questions?

Here are some related questions to expand on this concept:
1. What is the process of solving a system of equations using matrices?
2. How can quadratic regression be used for more than three points?
3. What are the characteristics of a parabola based on the signs of \(a\), \(b\), and \(c\)?
4. How does one find the vertex of a quadratic equation like this?
5. How would changing one of the points affect the resulting quadratic equation?

**Tip**: When solving quadratic equations, always verify your solution by plugging the values of \(x\) and \(y\) back into the equation to ensure consistency.

Use quadratic regression to find the equation for the parabola going through these 3 points: (-13, -126), (-7, 24), and (3, -46)

Learn how to use quadratic regression to find the equation of a parabola passing through the points (-13, -126), (-7, 24), and (3, -46). This tutorial covers key algebraic concepts, including solving quadratic equations and systems of equations. Suitable for high school students in Grades 9-12.

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