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.

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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