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Write a function that would match the graph 

Looking at the graph provided, the curve resembles a quartic function, specifically a polynomial of degree 4 due to its two turning points and symmetry. It appears to have the following characteristics:

1. It crosses the x-axis at \( x = -6, 0, 4 \), which suggests these are the roots of the function.
2. The function opens downwards, meaning the leading coefficient is negative.
3. The y-intercept appears to be 0, which also confirms that \( f(0) = 0 \).

Based on these points, we can express the function as:

\[
f(x) = a(x+6)x(x-4)
\]

where \(a\) is a negative constant that determines the vertical stretch or compression. To find \(a\), we can estimate the y-value at a specific x-coordinate, say \( x = 2 \), and substitute into the equation.

Would you like me to estimate \(a\) and give you the full function or further details about the graph analysis? 

Here are 5 questions to extend this topic:

1. What other types of graphs can be represented by quartic functions?
2. How can the leading coefficient affect the shape of the graph?
3. How do we use the roots of a polynomial to write the equation?
4. What are turning points, and how do they relate to the degree of a function?
5. How can you use transformations (shifts, stretches) to modify basic polynomial graphs?

**Tip:** Roots and symmetry often provide critical clues in determining the equation of a polynomial function!

Write a function in any form that would match the graph shown below.

Learn how to derive a quartic function based on its graph. This problem involves analyzing the roots, turning points, and leading coefficient of a polynomial. Suitable for high school students exploring polynomial functions in grades 10-12.

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