The problem involves analyzing a polynomial function $f(x)$ based on its graph and filling out the information regarding the degree, leading coefficient, number of distinct real zeros, and relative extremes.

Let's break this down step-by-step based on the graph shown:

1. Degree of $f(x)$:

   • The graph of the polynomial function crosses the x-axis 3 times and has a total of 4 turning points (which suggests it's a 5th-degree polynomial).
   • A polynomial's degree is equal to the number of its turning points plus 1 for odd-degree polynomials. Therefore, the degree of $f(x)$ is 5.

2. Leading Coefficient:

   • The end behavior of the graph can help identify the sign of the leading coefficient. Since the graph starts in the negative y-direction and ends in the positive y-direction (falling on the left and rising on the right), this indicates that the leading coefficient is positive.

3. Distinct Real Zeros:

   • The graph crosses the x-axis in three distinct places, indicating that there are 3 distinct real zeros.

4. Relative Extremes:

   • There are turning points where the graph changes direction (from increasing to decreasing or vice versa). In this case, there are 4 relative extremes (local maxima or minima).

Filling in the blanks:

• The degree of $f(x)$ is 5.
• The leading coefficient is positive.
• There are 3 distinct real zeros.
• There are 4 relative extremes.

Would you like further details on this, or do you have any specific questions?

Related Questions:

1. How can you determine the degree of a polynomial from its graph?
2. What does the leading coefficient of a polynomial tell you about the graph's end behavior?
3. How do you count the number of real zeros from a graph?
4. What is the difference between a local maximum and a local minimum on a polynomial graph?
5. Can the degree of a polynomial be even if its graph crosses the x-axis?

Tip:

The number of real zeros of a polynomial is equal to the number of times it intersects the x-axis, and the degree of the polynomial provides an upper bound for the total number of zeros (real and complex combined).