Let’s analyze the end behavior of each polynomial function from the image.

(a) $f(x) = -4x^3 - 4x^2 - 2x + 4$

• The degree of the polynomial is 3 (odd), and the leading coefficient is $-4$ (negative).
• For odd-degree polynomials:
  • If the leading coefficient is negative: The graph falls to the left and rises to the right.

Correct answer: Falls to the left and rises to the right.

(b) $f(x) = 4x^6 - 5x^5 + 8x^4 + 9x^2$

• The degree of the polynomial is 6 (even), and the leading coefficient is $4$ (positive).
• For even-degree polynomials:
  • If the leading coefficient is positive: The graph rises to the left and rises to the right.

Correct answer: Rises to the left and rises to the right.

(c) $f(x) = 3(x - 1)(x + 5)^2$

• Expand $f(x)$: The degree of the polynomial is $1 + 2 = 3$ (odd), and the leading coefficient is $3$ (positive).
• For odd-degree polynomials:
  • If the leading coefficient is positive: The graph falls to the left and rises to the right.

Correct answer: Falls to the left and rises to the right.

Summary of Answers:

1. (a) Falls to the left and rises to the right.
2. (b) Rises to the left and rises to the right.
3. (c) Falls to the left and rises to the right.

Let me know if you'd like more details about the reasoning behind these behaviors!

Relative Questions:

1. How does the degree of a polynomial affect its end behavior?
2. Why does the sign of the leading coefficient matter in determining the graph's behavior?
3. Can a polynomial's end behavior ever change if the degree remains constant?
4. How do multiplicities of roots affect the graph's shape near the roots?
5. What happens to the graph of a polynomial when the leading coefficient is zero?

Tip:

For quick analysis of end behavior, focus solely on the highest-degree term—it dictates the long-term trend of the polynomial.