https://file.aiquickdraw.com/mathbot/file/1726260615341wt12qvlg.jpg

The image shows a challenge asking to plot a parabola through three given points: 

- \((-9, 4)\)
- \((-6, 1)\)
- \((-3, 4)\)

To find the equation of the parabola that passes through these points, we can assume a general quadratic equation:

\[
y = ax^2 + bx + c
\]

We substitute the three points into this equation to form a system of three equations to solve for \(a\), \(b\), and \(c\).

1. For \((-9, 4)\):
   \[
   4 = a(-9)^2 + b(-9) + c
   \]
   \[
   4 = 81a - 9b + c
   \]

2. For \((-6, 1)\):
   \[
   1 = a(-6)^2 + b(-6) + c
   \]
   \[
   1 = 36a - 6b + c
   \]

3. For \((-3, 4)\):
   \[
   4 = a(-3)^2 + b(-3) + c
   \]
   \[
   4 = 9a - 3b + c
   \]

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

- \(a = \frac{1}{3}\)
- \(b = 4\)
- \(c = 13\)

Thus, the equation of the parabola passing through the points \((-9, 4)\), \((-6, 1)\), and \((-3, 4)\) is:

\[
y = \frac{1}{3}x^2 + 4x + 13
\]

Would you like further explanation on solving these types of systems, or any additional details?

Here are 5 related questions to consider:
1. How can you verify if a quadratic equation fits multiple points?
2. What happens to the parabola if you change the sign of \(a\)?
3. Can you generalize the process of finding a parabola for any set of three points?
4. How does the vertex form of a parabola relate to its standard form?
5. How would you solve this system graphically instead of algebraically?

**Tip:** When working with systems of equations, always check for consistency by substituting the solutions back into the original equations.

Plot a parabola through the points (-9, 4), (-3, 4), and (-6, 1).

Learn how to plot a parabola through points (-9, 4), (-3, 4), and (-6, 1) using the quadratic equation y = ax^2 + bx + c. This solution involves solving a system of equations to find the coefficients and graphing the resulting parabola. Suitable for students in Grades 9-12.

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