We are asked to use the leading coefficient test to determine the end behavior of the given polynomial function:

$f(x) = 7x^7 - 6x^4 + 8x^2 + 3$

Step-by-Step Solution:

1. Identify the leading term:
   The term with the highest degree is the leading term. In this polynomial, the leading term is $7x^7$.

2. Degree of the polynomial:
   The degree is the exponent of the leading term, which is $7$ (odd degree).

3. Leading coefficient:
   The leading coefficient is the coefficient of the leading term, which is $7$ (positive).

4. End Behavior Rules for Polynomials:

   • Odd degree, positive leading coefficient:
     • As $x \to -\infty$ (left side of the graph), $f(x) \to -\infty$ (falls left).
     • As $x \to +\infty$ (right side of the graph), $f(x) \to +\infty$ (rises right).

Thus, based on the leading coefficient test:

• End behavior: Falls left, rises right.

Answer:

The correct option is C: Falls left & rises right.

Would you like further details on any part of this solution?

Here are 5 related questions to explore further:

1. How would the end behavior change if the leading coefficient were negative?
2. What would the end behavior be for an even-degree polynomial with a positive leading coefficient?
3. How do the other terms of the polynomial affect the graph besides the end behavior?
4. Can you sketch the graph of this polynomial based on the end behavior and turning points?
5. What is the significance of the other terms in the polynomial for local behavior?

Tip: For polynomials, only the leading term determines the end behavior!