Problem 85 - Algebra and Trigonometry

Question

Explain the relationship between the degree of a polynomial function and the number of turning points on its graph.

Step-by-Step Solution

Answer

The degree of a polynomial function influences the maximum number of turning points in its graph. A polynomial of degree \(n\) can have up to \(n-1\) turning points.

Step 1: Introduction

A polynomial function is an expression of the form \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2 + a_1x + a_0\), where each \(a_i\) is a constant, \(n\) is a nonnegative integer, and \(a_n\) is not zero. The highest power in the polynomial is called the degree of the polynomial.

Step 2: Turning points

Turning points refer to the points on a graph of a function where the graph changes direction, i.e., it shifts from increasing to decreasing or decreasing to increasing.

Step 3: Relationship

The degree of a polynomial function majorly influences the number of turning points on its graph. For a polynomial of degree \(n\), the maximum number of turning points the graph will have is \(n-1\). This is, however, an upper limit, the graph could have fewer turning points as well.

Step 4: Example

For instance, if the degree of the polynomial function is 3 (i.e., a cubic function), then the maximum number of turning points the function's graph can have is \(3-1 = 2\).