To determine the end behavior of the function $f(x) = -x^4 + 4x^2$ using the Leading Coefficient Test, we focus on the term with the highest degree, as it has the greatest impact on the behavior of the graph as $x$ approaches positive or negative infinity.

Steps to Analyze End Behavior:

1. Identify the Leading Term: In $f(x) = -x^4 + 4x^2$, the leading term (the term with the highest power of $x$) is $-x^4$.

2. Determine the Degree and Leading Coefficient:

   • The degree of $f(x)$ is 4, which is an even number.
   • The leading coefficient is $-1$, which is negative.

3. Use the Leading Coefficient Test: For a polynomial with an even degree and a negative leading coefficient:

   • As $x \to +\infty$ (x goes to positive infinity), $f(x) \to -\infty$.
   • As $x \to -\infty$ (x goes to negative infinity), $f(x) \to -\infty$.

Conclusion:

The graph of $f(x) = -x^4 + 4x^2$ will fall to negative infinity on both the left and the right ends. This means:

• As $x \to +\infty$, $f(x) \to -\infty$.
• As $x \to -\infty$, $f(x) \to -\infty$.

Would you like more details on how this impacts graph shape, or have any other questions?

---

Here are some related questions:

1. How would the end behavior change if the leading coefficient were positive?
2. What would the end behavior be for an odd-degree polynomial with a negative leading coefficient?
3. How does the presence of other terms like $4x^2$ affect the overall shape, though not the end behavior?
4. What are the possible turning points for a polynomial of degree 4?
5. Can the Leading Coefficient Test be used for non-polynomial functions?

Tip: The end behavior for any polynomial function depends solely on its highest-degree term!