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.