Let's break down both questions from the image.

Question 9:

The graph of a polynomial function $f$ is shown. Which statement is true about the end behavior of the polynomial function?

Analysis:
End behavior refers to what happens to the function as $x$ approaches infinity or negative infinity. Without the graph, we must consider typical polynomial end behaviors.

For high-degree polynomials:

• The sign of the leading term (highest degree term) determines the end behavior.
• For an odd-degree polynomial, the ends of the graph will point in opposite directions.
• For an even-degree polynomial, both ends will point in the same direction.

The answer will depend on whether the leading coefficient is positive or negative and whether the degree is even or odd.

Question 10:

State the degree and end behavior of $f(x) = 5 + 7x - 9x^2 + 4x^3$. Explain or show your reasoning.

Analysis:

• The degree of a polynomial is the highest power of $x$, which in this case is 3 (from the term $4x^3$).
• Since the degree is odd (3), the ends of the graph will point in opposite directions.
• The leading term, $4x^3$, has a positive coefficient, so as $x \to \infty$, $f(x) \to \infty$, and as $x \to -\infty$, $f(x) \to -\infty$.

Therefore, the degree of the polynomial is 3, and the end behavior is:

• As $x \to \infty$, $f(x) \to \infty$.
• As $x \to -\infty$, $f(x) \to -\infty$.

Would you like more detailed explanations or further clarification? Here are some related questions to expand your understanding:

1. How does the degree of a polynomial affect its end behavior?
2. What role does the leading coefficient play in determining end behavior?
3. How does the graph of an odd-degree polynomial differ from an even-degree polynomial?
4. Can you determine end behavior from a polynomial's factored form?
5. What happens to a polynomial's graph near its roots compared to its end behavior?

Tip: Always check the sign and degree of the leading term in a polynomial to predict its end behavior.