Problem 63 - College Algebra

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

A polynomial function of degree \(n\) has at most \(n-1\) turning points on its graph.

Step 1: Definition of a Polynomial Function

A polynomial function is a function that can be expressed in the form \(f(x) = a_n x^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_2, a_1, a_0\) are constants, and \(n\) is a nonnegative integer.

Step 2: Degree of a Polynomial

The degree of a polynomial function is the highest power of \(x\) in its expression. It is represented by the \(n\) in the expression given above. For example, in the polynomial \(f(x) = 2x^3 - 7x^2 + 2x + 1\), the degree is 3.

Step 3: Turning Points on a Graph

A turning point on a graph of a function is a point where the graph changes from rising to falling, or falling to rising. These are the local maximums and minimums of the function.

Step 4: Relationship between Degree and Turning Points

The relationship between the degree of a polynomial and the number of turning points on its graph is that a polynomial function of degree \(n\) has at most \(n-1\) turning points. This follows from the fact that each turning point corresponds to a factor of the polynomial, and a polynomial of degree \(n\) can have at most \(n\) factors.