https://file.aiquickdraw.com/mathbot/file/17272943232836im43wfq.jpg

The polynomial function given in the image is:

\[
f(x) = -\frac{1}{4}x^4 + 3x^2
\]

This is a quartic function (degree 4) with a negative leading coefficient, which suggests that the ends of the graph will point downward (like an upside-down "W" shape). Let's analyze the graph and match it with one of the four displayed options:

1. **Shape**: Since the highest degree term has a negative coefficient and the degree is even (quartic), the ends of the graph will go down as \( x \to \infty \) and \( x \to -\infty \). Additionally, the \( 3x^2 \) term suggests the graph will have a parabolic shape in the middle with some turning points.

2. **Critical Points**: The graph should have multiple turning points because of the quartic nature of the function. The graph will likely exhibit symmetry around the y-axis because there are only even powers of \(x\), implying that \( f(x) = f(-x) \).

### Now, looking at the graphs:

- The first graph (top left) has a shape similar to an upside-down "W", matching the description of a negative quartic function.
- The second graph (top right) shows a function with different behavior (its ends are not both down).
- The third graph (bottom left) is not symmetric around the y-axis.
- The fourth graph (bottom right) looks like a parabola, which is not a quartic function.

### Conclusion:
The correct graph corresponding to the function \( f(x) = -\frac{1}{4}x^4 + 3x^2 \) is the **first one (top left)**.

Would you like more details or have any other questions?

Here are 5 related questions:
1. What are the general properties of quartic functions?
2. How does the leading coefficient affect the end behavior of a polynomial?
3. What is the significance of symmetry in polynomial graphs?
4. How can we find turning points of a quartic function analytically?
5. What are the possible inflection points of a quartic polynomial?

**Tip**: Always look at the leading term of a polynomial to predict the overall shape of its graph!

Match the polynomial function f(x) = -(1/4)x^4 + 3x^2 with its graph.

Explore how to match the quartic polynomial function f(x) = -(1/4)x^4 + 3x^2 with its corresponding graph. Learn key properties such as end behavior and symmetry for students in Grades 10-12.

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