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!