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Graph the polynomial functions. a.) y = -2x(x +4)²(x - 3)

Graph the polynomial functions. a.) y = -2x(x +4)²(x - 3)

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Graph the polynomial function.

**Graph the Polynomial Functions** 1. **Problem Statement:** a) \( y = -2x(x + 4)^2(x - 3) \) **Explanation:** This is a polynomial function with four terms. The function is expressed in factored form, which allows us to easily identify the roots and the behavior of the graph at each root. - **Roots of the Polynomial:** - \( x = 0 \) - \( x = -4 \) (with a multiplicity of 2) - \( x = 3 \) - **Behavior Near the Roots:** - At \( x = 0 \), the graph crosses the x-axis. - At \( x = -4 \), due to the multiplicity of 2, the graph touches the x-axis and turns around, forming a parabolic behavior. - At \( x = 3 \), the graph crosses the x-axis. - **Leading Coefficient and End Behavior:** - The leading coefficient is -2, which is negative. This indicates that the graph will start with a positive end as \( x \) approaches negative infinity and will end with a negative value as \( x \) approaches positive infinity. Understanding these characteristics allows for a rough sketch of the graph. The graph has three x-intercepts, and its shape will be influenced by both the multiplicity of the roots and the negative leading coefficient.
Transcribed Image Text:**Graph the Polynomial Functions** 1. **Problem Statement:** a) \( y = -2x(x + 4)^2(x - 3) \) **Explanation:** This is a polynomial function with four terms. The function is expressed in factored form, which allows us to easily identify the roots and the behavior of the graph at each root. - **Roots of the Polynomial:** - \( x = 0 \) - \( x = -4 \) (with a multiplicity of 2) - \( x = 3 \) - **Behavior Near the Roots:** - At \( x = 0 \), the graph crosses the x-axis. - At \( x = -4 \), due to the multiplicity of 2, the graph touches the x-axis and turns around, forming a parabolic behavior. - At \( x = 3 \), the graph crosses the x-axis. - **Leading Coefficient and End Behavior:** - The leading coefficient is -2, which is negative. This indicates that the graph will start with a positive end as \( x \) approaches negative infinity and will end with a negative value as \( x \) approaches positive infinity. Understanding these characteristics allows for a rough sketch of the graph. The graph has three x-intercepts, and its shape will be influenced by both the multiplicity of the roots and the negative leading coefficient.
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