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Chapter 2: Problem 102

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm graphing a fourth-degree polynomial function with four turning points.

Short Answer

Expert verified
The given statement does not make sense because a fourth-degree polynomial function can have at most three turning points, not four.

Step by step solution

01

Understanding Polynomial Functions and Their Graphs

A polynomial function is of the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where \(a_n\) is non-zero. The degree of the polynomial is the highest power \(n\) of \(x\). In the case of a graph, each time the graph changes direction, this is described as a turning point.
02

Understanding the Relation Between Degree and Turning Points

The degree of the polynomial function limits the maximum number of turning points the function can have. Generally, a polynomial function of degree \(n\) can have up to \(n - 1\) turning points. Hence, a fourth-degree polynomial function may have up to three turning points.
03

Analyzing the Given Statement

The statement posits that 'I'm graphing a fourth-degree polynomial function with four turning points.' Given the understanding from the earlier steps, this does not make sense. A fourth-degree polynomial function can have at most three turning points, not four.

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