Question

Sketch the function. $f(x)=\frac{1}{2}(x-2)(x+1)(x+3)$

   Sketch the function.
 $f(x)=\frac{1}{2}(x-2)(x+1)(x+3)$

College Algebra Essentials

College Algebra Essentials

Julie Miller 1st Edition

Chapter 3, Problem 65

Instant Answer

Solved by Expert Nick Johnson

08/08/2021

Step 1

Since the highest power of x in the function is 3, we know that the function is a cubic function. The general shape of a cubic function is down to the left and up to the right. Show more…

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Sketch the function. $f(x)=\frac{1}{2}(x-2)(x+1)(x+3)$

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Key Concepts

Polynomial Functions

A polynomial function is an expression of one or more terms that consist of a coefficient multiplied by a variable raised to a nonnegative integer power. The overall behavior of the graph, including its turning points and end behavior, is largely determined by the degree (highest power) of the polynomial.

Factored Form and Zeros

Expressing a polynomial in factored form allows for easy identification of its zeros (roots), which are the x-values for which the function equals zero. These zeros indicate where the graph crosses or touches the x-axis, and they are crucial for sketching the function accurately.

Multiplicity of Roots

The multiplicity of a root refers to the number of times a particular zero appears in the factored form. The multiplicity affects the graph's behavior at that zero; for example, roots with an odd multiplicity typically cross the x-axis, while those with an even multiplicity usually just touch the axis and turn around.

End Behavior Analysis

End behavior describes how a polynomial function behaves as the input variable approaches positive or negative infinity. This behavior is determined by the leading term (the term with the highest power) of the polynomial and provides guidance on how the graph extends far to the left and right.

Intercepts

Intercepts are the points where the graph meets the coordinate axes. The x-intercepts are found by solving for the zeros from the factored form, while the y-intercept is found by evaluating the function at x = 0. These points help anchor the graph and guide its overall shape.

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