https://file.aiquickdraw.com/mathbot/file/17267623320802o2dazmr.jpg

Let's analyze the graph provided.

The curve looks like a cubic function, as it has two turning points (maximum and minimum). Here's the step-by-step breakdown:

### Identifying the Roots (Zeros):
The graph intersects the x-axis at three points. These points are the roots (or zeros) of the function.
From the graph, the approximate x-intercepts (roots) appear to be at:
- \( x = -2 \)
- \( x = 0 \)
- \( x = 2 \)

### Factored Form of the Polynomial:
The factored form of a cubic function with these roots can be written as:
\[
y = a(x + 2)(x)(x - 2)
\]
where \(a\) is the leading coefficient.

### Determine the Leading Coefficient:
We are told to assume that the leading coefficient \( a = \pm 1 \). From the graph, the function appears to be increasing on the right, so we choose \( a = 1 \).

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

### Simplification:
Now, simplifying the factored form:
\[
y = (x + 2)(x)(x - 2) = x(x^2 - 4) = x^3 - 4x
\]

So, the equation of the curve is:
\[
y = x^3 - 4x
\]

Would you like further details or explanations on any of the steps? Here are some related questions to expand your understanding:

1. How can you find the turning points of a cubic function?
2. What role does the leading coefficient play in determining the direction of the graph?
3. Can you graphically determine whether a function is odd or even?
4. How does changing the value of \(a\) (the leading coefficient) affect the shape of the cubic graph?
5. How do the x-intercepts of a polynomial relate to its factored form?

**Tip**: For polynomials, the degree of the function is equal to the number of roots (including multiplicities), and the shape of the graph gives clues about the leading coefficient.

Determine an equation for the pictured graph. Write your answer in factored form and assume the leading coefficient is ±1.

Learn how to determine the factored form of a cubic function based on its graph. This problem involves identifying x-intercepts and applying the root theorem to write the polynomial's equation. Suitable for students in Grades 10-12.

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