https://file.aiquickdraw.com/mathbot/file/172677353872421iu3uyi.jpg

The graph shown in the image is of a cubic function due to its S-shaped nature, which suggests that its highest degree term is \( x^3 \). Based on this, we can eliminate the options that involve quadratic equations.

The equation of the graph must also capture the intercepts. From the graph, it passes through the x-axis at points \( x = -2 \), \( x = 0 \), and \( x = 2 \). These x-intercepts suggest that the function has factors of \( (x+2), x, (x-2) \), which leads to an expression like \( y = x(x - 2)(x + 2) \), simplifying to:

\[
y = x(x^2 - 4) = x^3 - 4x
\]

This matches option D: \( y = x^3 - 4x \).

Thus, the correct answer is **D**.

Would you like any further details on the process or graph interpretation? 

Here are five related questions to deepen your understanding:
1. How can we distinguish between quadratic and cubic functions from their graphs?
2. What are the key features of a cubic function graph?
3. How can we derive the equation of a polynomial function from its graph?
4. What role do intercepts play in determining the factors of a function?
5. Can we determine if a cubic function has turning points by looking at the graph?

**Tip:** For cubic functions, look for the characteristic S-shape and check for the number of x-intercepts to identify potential factors.

Which equation matches the cubic graph provided?

Analyze a cubic function graph and determine its equation by factoring the x-intercepts. This problem involves understanding the shape of cubic functions, the Factor Theorem, and graph interpretation for polynomial equations. Suitable for high school students in Grades 9-12.

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