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This image shows a graph where you're asked to plot a parabola through three points: one at the vertex marked at \( (4, -2) \), and two other points at \( (0, 8) \) and \( (8, 0) \).

### Steps to find the equation of the parabola:

1. **General form of the parabola**: The equation of a parabola can be written as:
   \[
   y = a(x - h)^2 + k
   \]
   where \( (h, k) \) is the vertex of the parabola. Given that the vertex is \( (4, -2) \), the equation becomes:
   \[
   y = a(x - 4)^2 - 2
   \]
   
2. **Using one of the points to solve for \( a \)**:
   Let's use the point \( (0, 8) \) to find \( a \):
   \[
   8 = a(0 - 4)^2 - 2
   \]
   Simplifying:
   \[
   8 = 16a - 2
   \]
   \[
   10 = 16a
   \]
   \[
   a = \frac{10}{16} = \frac{5}{8}
   \]

3. **Equation of the parabola**:
   Now that we know \( a = \frac{5}{8} \), the equation of the parabola is:
   \[
   y = \frac{5}{8}(x - 4)^2 - 2
   \]

4. **Verification**:
   To verify, substitute \( (8, 0) \) into the equation:
   \[
   0 = \frac{5}{8}(8 - 4)^2 - 2
   \]
   Simplifying:
   \[
   0 = \frac{5}{8}(16) - 2
   \]
   \[
   0 = 10 - 2
   \]
   \[
   0 = 0
   \]
   This confirms that the equation is correct.

So, the parabola that passes through the given points has the equation:
\[
y = \frac{5}{8}(x - 4)^2 - 2
\]

Would you like any further details or explanations on this solution?

Here are 5 related questions you might want to explore:
1. How do you find the axis of symmetry for a parabola?
2. What changes in the graph of a parabola if the value of \( a \) is negative?
3. Can you find the focus and directrix of this parabola?
4. How can you graph parabolas in standard form without knowing the vertex?
5. How do parabolas differ from other conic sections, like ellipses or hyperbolas?

**Tip**: When solving for the equation of a parabola, always use the vertex form \( y = a(x-h)^2 + k \), which simplifies solving for the parameters given vertex and a few points.

Plot a parabola through the points (4, -2), (0, 8), and (8, 0). The vertex is given at (4, -2).

Learn how to plot a parabola through points (0, 8), (8, 0), and (4, -2) with the vertex form equation. This example explains how to solve for 'a' using the given points, leading to the final equation of the parabola. Suitable for Grades 9-11 students.

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