Create a question that determine an equation of a parabola in factored form graphically on grid. Include the graph and solution.

Here is a graph of a parabola with x-intercepts at \(x = -3\) and \(x = 2\), and the vertex at \((-\frac{1}{2}, -4)\). Based on this, we can determine the equation of the parabola in factored form.

### Finding the Equation:
The general factored form of a parabola is:
\[
y = a(x - r_1)(x - r_2)
\]
where \(r_1\) and \(r_2\) are the x-intercepts of the parabola.

For this graph:
- \(r_1 = -3\)
- \(r_2 = 2\)

So, the equation becomes:
\[
y = a(x + 3)(x - 2)
\]

Now, to find the value of \(a\), we use the vertex \((-\frac{1}{2}, -4)\).

Substitute \(x = -\frac{1}{2}\) and \(y = -4\) into the equation:
\[
-4 = a\left(-\frac{1}{2} + 3\right)\left(-\frac{1}{2} - 2\right)
\]
\[
-4 = a\left(\frac{5}{2}\right)\left(-\frac{5}{2}\right)
\]
\[
-4 = a \left(-\frac{25}{4}\right)
\]
\[
a = \frac{-4}{-\frac{25}{4}} = \frac{16}{25}
\]

Thus, the equation of the parabola is:
\[
y = \frac{16}{25}(x + 3)(x - 2)
\]

### Summary:
The parabola has the factored form equation \( y = \frac{16}{25}(x + 3)(x - 2) \).

Do you want any further details on this problem or its steps?

### Questions:
1. How do you convert from vertex form to factored form?
2. How can you identify the vertex and intercepts from the equation of a parabola?
3. What other methods can you use to find the equation of a parabola from its graph?
4. How does changing the value of \(a\) affect the shape of the parabola?
5. Can you determine the axis of symmetry from the factored form of the equation?

### Tip:
Remember that the coefficient \(a\) controls the "stretch" of the parabola; the larger the value of \(|a|\), the narrower the parabola.

Create a question that determines the equation of a parabola in factored form graphically on a grid. Include the graph and solution.

Explore how to find the equation of a parabola in factored form using a graph. This problem involves key algebraic concepts, including the factored form of quadratic equations and vertex theorem, suitable for students in Grades 9-12.

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