Let’s break this down and address the questions one by one based on the uploaded worksheet.

Part 1

1. Graph 1:

• (a) End behavior:
  For the given graph, the ends point in the same direction (upward), which indicates the function behaves like an even-degree polynomial. Specifically:

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

• (b) Even or odd degree?
  Since both ends point in the same direction, this is the graph of an even-degree function.

• (c) Leading coefficient:
  The leading coefficient is positive because both ends point upward.

2. Graph 2:

• (a) End behavior:
  The ends point in opposite directions, which indicates the function behaves like an odd-degree polynomial. Specifically:

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

• (b) Even or odd degree?
  Since the ends point in opposite directions, this is the graph of an odd-degree function.

• (c) Leading coefficient:
  The leading coefficient is negative because the right end of the graph points downward.

Part 2

3. End behavior of a 14th-degree polynomial with a positive leading coefficient:

• A 14th-degree polynomial is an even-degree polynomial.
  • As $x \to \infty, f(x) \to \infty$.
  • As $x \to -\infty, f(x) \to \infty$.
    Both ends point upward because the leading coefficient is positive.

4. End behavior of a 9th-degree polynomial with a negative leading coefficient:

• A 9th-degree polynomial is an odd-degree polynomial.
  • As $x \to \infty, f(x) \to -\infty$.
  • As $x \to -\infty, f(x) \to \infty$.
    The right end points downward, and the left end points upward due to the negative leading coefficient.

Part 3

5. Key characteristics of $f(x) = -x^3 + 6x^2 - 10x + 4$:

Let’s outline the features step by step:

1. Domain:
   The domain of a polynomial is all real numbers:
   $\text{Domain: } (-\infty, \infty)$

2. Range:
   This depends on the graph but would typically span all real numbers for a cubic function:
   $\text{Range: } (-\infty, \infty)$

3. Relative Maximum(s):
   Use a graphing calculator or derivative test to find local maxima (rounded to the nearest hundredth).

4. Relative Minimum(s):
   Similarly, use a graphing calculator or derivative test to find local minima.

5. End Behavior:
   Since the leading term is $-x^3$:

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

6. Increasing Intervals:
   Identify intervals where the slope (derivative) is positive.

7. Decreasing Intervals:
   Identify intervals where the slope (derivative) is negative.

Would you like me to calculate the exact turning points or solve anything specific from this? Let me know!

Related Questions

1. How do you determine the degree of a polynomial from its graph?
2. What are the general rules for the end behavior of even- and odd-degree polynomials?
3. How do positive or negative leading coefficients affect the graph?
4. What is the process for finding increasing and decreasing intervals for a function?
5. How do you compute relative maxima and minima for polynomials?

Tip

When determining the end behavior of a polynomial, focus only on the highest-degree term, as it dominates for large values of $x$.