https://file.aiquickdraw.com/mathbot/file/1726545316421c1xqp7qq.jpg

The parabola in the image is shown with its x-intercepts at \((-2, 0)\) and \((2, 0)\), and its vertex at \((1, -2)\).

To write the equation of a parabola in intercept (factored) form, we use:

\[
y = a(x - p)(x - q)
\]

Where \( p \) and \( q \) are the x-intercepts. Here, the intercepts are \( p = -2 \) and \( q = 2 \). Substituting these values, we get:

\[
y = a(x + 2)(x - 2)
\]

Now, we use the vertex \((1, -2)\) to find the value of \(a\). Substituting \(x = 1\) and \(y = -2\) into the equation:

\[
-2 = a(1 + 2)(1 - 2)
\]

Simplifying:

\[
-2 = a(3)(-1)
\]

\[
-2 = -3a
\]

Solving for \(a\):

\[
a = \frac{-2}{-3} = \frac{2}{3}
\]

Thus, the equation of the parabola is:

\[
y = \frac{2}{3}(x + 2)(x - 2)
\]

This is the required equation in intercept form.

Would you like further clarification or details?

Here are 5 related questions to enhance your understanding:
1. What does the value of \(a\) represent in the equation of a parabola?
2. How do we identify the vertex of a parabola from its graph?
3. What are the key differences between vertex form and intercept form of a parabola?
4. How would the equation change if the parabola opened downward?
5. How can you find the axis of symmetry for a parabola from its intercepts?

**Tip**: The value of \(a\) controls the "stretch" and direction of the parabola—if \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.

Write an equation of the parabola in intercept form using the given x-intercepts (-2, 0) and (2, 0) and the vertex (1, -2).

Learn how to find the equation of a parabola using the intercept form with given x-intercepts (-2, 0) and (2, 0) and the vertex (1, -2). This problem involves the application of algebra and quadratic equations, suitable for students in Grades 9-11.

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