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.