Let's analyze the given function $f(x) = 8 - 2x + \frac{3}{4}x^4$.

(a) Degree and Leading Coefficient:

• The degree of a polynomial is the highest power of $x$ in the function. In this case, the term $\frac{3}{4}x^4$ has the highest power of $x$, which is 4. Therefore, the degree of $f(x)$ is 4.
• The leading coefficient is the coefficient of the term with the highest degree. For the term $\frac{3}{4}x^4$, the coefficient is $\frac{3}{4}$. Therefore, the leading coefficient is $\frac{3}{4}$.

(b) End Behavior:

The end behavior of a polynomial depends on the degree and the leading coefficient:

• For even-degree polynomials, if the leading coefficient is positive, the graph rises to both the left and the right.
• If the leading coefficient is negative, the graph falls to both the left and right.

Since the degree is 4 (even) and the leading coefficient $\frac{3}{4}$ is positive, the graph of $f(x)$ rises both to the left and the right.

Therefore, the correct answer for part (b) is B: "The graph of $f$ rises both to the left and to the right."

Would you like more details or have any other questions? Here are five related questions you might consider:

1. What would happen to the graph's end behavior if the leading coefficient were negative?
2. How does the degree of a function affect the shape of its graph?
3. Can you explain how odd-degree polynomials behave differently in terms of end behavior?
4. How do we determine the y-intercept from a polynomial function like $f(x)$?
5. Can a polynomial function have more than one turning point?

Tip: The end behavior of a polynomial can be easily predicted by focusing on the term with the highest degree, as lower-degree terms have little influence on the graph at extreme values of $x$.