https://file.aiquickdraw.com/mathbot/file/1728275343690gy9ynvti.jpg

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 \( x \).

Complete parts (a) and (b) for f(x) = 8 - 2x + (3/4)x^4. (a) State the degree and leading coefficient of f. (b) State the end behavior of the graph of f.

Learn how to determine the degree and leading coefficient of the polynomial function f(x) = 8 - 2x + (3/4)x^4. Explore how the degree and leading coefficient impact the graph's end behavior. Suitable for high school students (Grades 10-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation