To match the polynomial function $f(x) = -\frac{1}{4}x^4 + 3x^2$ with its graph, let's analyze the function step by step:

1. Degree and End Behavior:

• The degree of the polynomial is 4 (the highest exponent is 4), which means it is a quartic function.
• Since the leading term has a negative coefficient $-\frac{1}{4}x^4$, the ends of the graph will point downward. As $x \to \infty$ or $x \to -\infty$, $f(x) \to -\infty$. This gives the graph a "W" shape, but inverted because of the negative leading term.

2. Turning Points:

• A quartic polynomial can have up to 3 turning points (local maxima and minima). The number of turning points helps determine the general shape of the graph.

3. Symmetry:

• The function is an even function because only even powers of $x$ are present (no odd powers of $x$). Thus, the graph is symmetric with respect to the y-axis.

4. Intercepts:

• y-intercept: Plugging $x = 0$ into the function, we get: $f(0) = -\frac{1}{4}(0)^4 + 3(0)^2 = 0$ So the graph passes through the origin $(0, 0)$.
• x-intercepts: To find the x-intercepts, solve $f(x) = 0$: $-\frac{1}{4}x^4 + 3x^2 = 0$ Factor the equation: $x^2\left(-\frac{1}{4}x^2 + 3\right) = 0$ The solutions are: $x^2 = 0 \quad \text{or} \quad -\frac{1}{4}x^2 + 3 = 0$ From $x^2 = 0$, we get $x = 0$. Solving $-\frac{1}{4}x^2 + 3 = 0$, we get: $-\frac{1}{4}x^2 = -3 \quad \Rightarrow \quad x^2 = 12 \quad \Rightarrow \quad x = \pm 2\sqrt{3}$ So the x-intercepts are $x = 0$ and $x = \pm 2\sqrt{3}$.

5. Behavior near the origin:

• Near the origin, the graph behaves like a parabola because the $3x^2$ term dominates. The $-\frac{1}{4}x^4$ term will flatten the graph away from the origin as $|x|$ increases.

Conclusion:

• The graph should be symmetric about the y-axis, pass through the origin, have three x-intercepts (including $\pm 2\sqrt{3}$), and exhibit "W"-like behavior (inverted) because of the negative quartic term.

Would you like me to generate a graph for this function or clarify any specific part?

Related Questions:

1. What are the critical points of $f(x)$ and how do we find them?
2. How does the graph change if we alter the coefficient of the quartic term?
3. Can you explain the end behavior of a general quartic function?
4. How do we calculate turning points using derivatives for $f(x)$?
5. What are the possible shapes of graphs for even-degree polynomials?

Tip:

The symmetry of a polynomial function can often be identified by checking whether all powers of $x$ are even (even function) or odd (odd function).