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.