https://file.aiquickdraw.com/mathbot/file/1727520431603np2wdq52.jpg

The task in the image is to find the equation of the parabola shown in the figure, in the form:

\[
y = a(x - x_1)(x - x_2)
\]

where \( (x_1, 0) \) and \( (x_2, 0) \) are the x-intercepts of the parabola. From the graph, it is clear that the parabola opens upwards and passes through the point \( (4, -4) \). We can proceed with the following steps:

### Step 1: Identify the x-intercepts
From the graph, it seems that the parabola touches the x-axis at \( x_1 = -4 \) and \( x_2 = 4 \). These are the roots of the parabola.

### Step 2: Use the general form \( y = a(x - x_1)(x - x_2) \)
Substitute \( x_1 = -4 \) and \( x_2 = 4 \) into the equation:

\[
y = a(x + 4)(x - 4)
\]

This simplifies to:

\[
y = a(x^2 - 16)
\]

### Step 3: Find the value of \( a \)
To determine \( a \), use the fact that the parabola passes through the point \( (0, -4) \). Substitute \( x = 0 \) and \( y = -4 \) into the equation:

\[
-4 = a(0^2 - 16)
\]
\[
-4 = -16a
\]
\[
a = \frac{-4}{-16} = \frac{1}{4}
\]

### Step 4: Write the final equation
Substitute \( a = \frac{1}{4} \) into the equation:

\[
y = \frac{1}{4}(x^2 - 16)
\]

Thus, the equation of the parabola is:

\[
y = \frac{1}{4}x^2 - 4
\]

Let me know if you need further details or clarifications!

### Related Questions:
1. How do you determine the vertex of a parabola given its equation in standard form?
2. What is the relationship between the coefficient \( a \) and the direction in which the parabola opens?
3. How do you find the focus of a parabola given its equation?
4. What are the properties of the parabola when \( a > 0 \) compared to when \( a < 0 \)?
5. How do the x-intercepts help in constructing the factored form of a quadratic equation?

### Tip:
To easily identify the shape of a parabola, remember that when \( a > 0 \), the parabola opens upwards, and when \( a < 0 \), it opens downwards.

Find an equation of the form y = a(x - x1)(x - x2) of the parabola shown in the figure.

Learn how to find the equation of a parabola in factored form given its x-intercepts and a point it passes through. This problem covers key concepts in algebra and quadratic functions, suitable for students in Grades 9-10.

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