https://file.aiquickdraw.com/mathbot/file/1728757133957dpbdj5l9.jpg

The graph in the image shows a parabola, and the task asks you to write the equation of this parabola in the form \( y = a(x - p)(x - q) \), where \( p \) and \( q \) are the x-intercepts, and \( a \) is a constant determined by another point on the graph.

### Key observations from the graph:
1. **X-Intercepts**: 
   - The parabola crosses the x-axis at \( x = -1 \) and \( x = 7 \). Therefore, \( p = -1 \) and \( q = 7 \).

2. **Another Point**:
   - A third point on the graph is given as \( (2, 2) \), which will help us determine the value of \( a \).

### Step-by-Step Process:
1. **Start with the general equation**:
   \[
   y = a(x - p)(x - q)
   \]
   Substituting \( p = -1 \) and \( q = 7 \):
   \[
   y = a(x + 1)(x - 7)
   \]

2. **Use the point \( (2, 2) \) to find \( a \)**:
   Substitute \( x = 2 \) and \( y = 2 \) into the equation:
   \[
   2 = a(2 + 1)(2 - 7)
   \]
   Simplify:
   \[
   2 = a(3)(-5)
   \]
   \[
   2 = -15a
   \]
   Solving for \( a \):
   \[
   a = \frac{2}{-15} = -\frac{2}{15}
   \]

3. **Final Equation**:
   The equation of the parabola is:
   \[
   y = -\frac{2}{15}(x + 1)(x - 7)
   \]

Would you like more details on any part of the process? Here are a few questions you could explore further:

1. How do we use x-intercepts to form the equation of a parabola?
2. Why do we need a third point to find the value of \( a \)?
3. What happens if we change the x-intercepts in the equation?
4. How can we verify the equation of a parabola using graphing?
5. How do parabolas behave when the value of \( a \) is positive vs negative?

**Tip**: Always double-check the signs of the x-intercepts when plugging them into the equation, as this is a common source of errors.

Write the equation of the parabola shown in the graph, given that the x-intercepts are (-1, 0) and (7, 0) and another point on the graph is (2, 2).

Learn how to write the quadratic equation of a parabola by using its x-intercepts and another point. This step-by-step guide explains how to apply the factored form of the quadratic equation and solve for the unknown 'a'. Suitable for Grades 9-10 algebra students.

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