The possible degrees of the function are 7, 8, and 9.

UpStudy Free Solution

Step 1: Identify the number of turning points - A turning point occurs where the graph changes direction from increasing to decreasing or vice versa.  - Count the number of turning points in the given graph: There are 6 turning points.

Step 2: Determine the relationship between turning points and the degree of the polynomial - A polynomial of degree \(n\) can have at most \(n- 1\) turning points. - Thus, if there are 6 turning points, the minimum degree of the polynomial must be \(n \geq 7\) (since \(7 - 1 = 6\)).

Step 3: Determine the possible degrees from the given options - The options are 4, 5, 6, 7, 8, and 9. - Based on our analysis, the degree must be at least 7. - Therefore, the possible degrees are 7, 8, and 9.

Key Concepts:

1. Turning Points: A turning point on the graph of a polynomial is a point where the graph changes direction from increasing to decreasing or decreasing to increasing. These points indicate local maxima or minima.

2. Degree of a Polynomial: The highest power of the variable in a polynomial expression gives the degree of the polynomial. The degree provides the general shape and maximum number of turning points of the polynomial.

Explanation:

- Identifying Turning Points: To find the number of turning points, observe the graph and count how many times it changes direction. Each change in direction corresponds to a turning point.

- Relationship Between Turning Points and Degree: For a polynomial of degree \(n\), the maximum number of turning points it can have is \(n- 1\). This relationship helps in determining the minimum possible degree of a polynomial when the number of turning points is known.

- Determining the Degree:

- If a polynomial has \(t\) turning points, the degree \(n\) must satisfy \(n- 1 \geq t\). Therefore, \(n\) must be at least \(t+ 1\).

- For example, if there are 5 turning points, the minimum degree \(n\) must be 6 (since \(6- 1 = 5\)).

- Analyzing Given Options: Once the minimum degree is established, compare it to the choices for the candidate degrees. For instance, if the options are 4, 5, 6, 7, 8, and 9, and the minimum degree is seven according to turning points, the candidate degrees are 7, 8, and 9.

These principles will enable us to analyze polynomial functions based on their graphs and to determine possible degrees when we observe characteristics such as turning points. Get the free app that knows how to solve your math problems—UpStudy Algebra Calculator. Get the UpStudy app, learn better with intuitive, well-explained tutorials and improve your performance in class. Learn better with UpStudy!