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Based on the image, we are analyzing the graphs of two polynomial functions, both of degree 5 (quintic functions), and the related questions about their behavior.

Let's go over each of the questions and details:

### Questions:
1. **What do you notice about the number of turning points of the quartic functions (numbers 1 and 2)? How about of quintic functions (numbers 3 to 5)?**
    - **Quartic functions** (degree 4) typically have a maximum of 3 turning points.
    - **Quintic functions** (degree 5) can have up to 4 turning points, as seen in the examples.

2. **From the given examples, do you think it is possible for the degree of a function to be less than the number of turning points?**
    - No, the number of turning points is always less than or equal to the degree of the polynomial minus 1. So, a function of degree \( n \) can have at most \( n-1 \) turning points.

3. **State the relation of the number of turning points of a function with its degree \( n \).**
    - For a polynomial of degree \( n \), the maximum number of turning points is \( n-1 \). This is because the derivative of a degree \( n \) polynomial is a polynomial of degree \( n-1 \), and turning points occur where the derivative is zero.

### Summary:
- Quintic functions (degree 5) can have up to 4 turning points.
- The number of turning points is always less than or equal to \( n-1 \), where \( n \) is the degree of the polynomial.

Would you like more details on any of the above answers, or help with the concepts involved?

Here are 5 related questions to expand on this:
1. What is the general shape of polynomials of degree 3 (cubic) compared to degree 4 (quartic)?
2. How can the number of real roots of a polynomial function be related to its turning points?
3. What determines whether a turning point is a local maximum or a local minimum?
4. How does the leading coefficient of a polynomial affect the end behavior of its graph?
5. Can a quintic function ever have no turning points at all? Why or why not?

**Tip:** The graph of the derivative of a polynomial can provide insights into the number and nature of the turning points of the polynomial itself.

What do you notice about the number of turning points of quartic and quintic functions? From the given examples, is it possible for the degree of a function to be less than the number of turning points? State the relation of the number of turning points with the degree of a polynomial.

Explore how to determine the number of turning points in quartic (degree 4) and quintic (degree 5) polynomial functions. Learn about the relationship between the degree of a polynomial and its maximum number of turning points, based on real-world examples.

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