Let's go over the three polynomial functions from the image and analyze their end behavior based on the degree and the leading coefficient:

(a) $f(x) = -5x^6 + 4x^5 - 3x^3 - 8x^2$

• Degree: The degree of this polynomial is 6 (the highest power of $x$).
• Leading coefficient: The leading coefficient is -5 (negative).
• End behavior: For a polynomial of even degree with a negative leading coefficient:
  • As $x \to \infty$, $f(x) \to -\infty$ (falls to the right).
  • As $x \to -\infty$, $f(x) \to -\infty$ (falls to the left).

So, the graph falls to the left and falls to the right.

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

• Degree: The degree of this polynomial is 3 (the highest power of $x$).
• Leading coefficient: The leading coefficient is -5 (negative).
• End behavior: For a polynomial of odd degree with a negative leading coefficient:
  • As $x \to \infty$, $f(x) \to -\infty$ (falls to the right).
  • As $x \to -\infty$, $f(x) \to \infty$ (rises to the left).

So, the graph rises to the left and falls to the right.

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

• Degree: First, we need to expand this polynomial. The degree of $x^2$ is 2, and $(3x - 5)^2$ is also degree 2, so the total degree is 4.
• Leading coefficient: The leading term after expansion will have a positive leading coefficient (the sign of $x^4$ will be positive).
• End behavior: For a polynomial of even degree with a positive leading coefficient:
  • As $x \to \infty$, $f(x) \to \infty$ (rises to the right).
  • As $x \to -\infty$, $f(x) \to \infty$ (rises to the left).

So, the graph rises to the left and rises to the right.

Let me know if you would like more details or have any questions! Here are some related questions for you to consider:

1. How do odd and even degrees affect end behavior?
2. What role does the leading coefficient play in determining the end behavior of a polynomial function?
3. Can the graph of a polynomial with even degree ever rise on one side and fall on the other?
4. How would the end behavior change if the leading coefficient of a polynomial function is positive?
5. How can you determine end behavior without fully expanding a polynomial like in (c)?

Tip: Always check the highest degree term and its sign (leading coefficient) to quickly determine the end behavior of polynomials!