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.