Construct a polynomial function that has the following graph. Write the final answer as an expanded polynomial (not in factored form). Show your work. y
Construct a polynomial function that has the following graph. Write the final answer as an expanded polynomial (not in factored form). Show your work. y
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter4: Polynomial And Rational Functions
Section4.2: Polynomial Functions
Problem 94E
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Question
![**Task: Construct a polynomial function that has the following graph. Write the final answer as an expanded polynomial (not in factored form). Show your work.**
**Graph Description:**
The graph shows a polynomial function on a coordinate plane with the x-axis and y-axis marked. Key features of the graph include:
- The graph crosses the x-axis at \( x = -2 \), \( x = 0 \), and \( x = 3 \), implying these are roots of the polynomial.
- It appears to change direction at these points, indicating possible turning points or roots with multiplicity.
- The overall shape of the graph is typical of a cubic polynomial, showing one peak and one trough.
**Key Points and Behavior:**
1. **Intercepts:**
- The graph intersects the x-axis at \( x = -2 \), \( x = 0 \), and \( x = 3 \).
2. **Shape:**
- The graph starts from the negative side of the y-axis, rises to a local maximum, decreases to a local minimum, and rises again, typical of a cubic function.
Given these points, the polynomial could be represented by a function of the form \( f(x) = a(x + 2)(x)(x - 3) \).
To fully determine the function, the coefficient \( a \) would need to be found, often through additional points or specific conditions which are not provided in the question description.
In expanded form, with \( a = 1 \) (assuming leading coefficient adjustments aren't specified):
\[ f(x) = (x + 2)x(x - 3) \]
\[ = (x^2 + 2x)(x - 3) \]
\[ = x^3 - 3x^2 + 2x^2 - 6x \]
\[ = x^3 - x^2 - 6x \]
This polynomial should match the given graph's shape and intercepts when plotted.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8f19db68-e430-43ee-998b-3e18aabae7e5%2F0ad303cb-5a54-48f8-b7a3-e413864e30d0%2F9onyyg8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Task: Construct a polynomial function that has the following graph. Write the final answer as an expanded polynomial (not in factored form). Show your work.**
**Graph Description:**
The graph shows a polynomial function on a coordinate plane with the x-axis and y-axis marked. Key features of the graph include:
- The graph crosses the x-axis at \( x = -2 \), \( x = 0 \), and \( x = 3 \), implying these are roots of the polynomial.
- It appears to change direction at these points, indicating possible turning points or roots with multiplicity.
- The overall shape of the graph is typical of a cubic polynomial, showing one peak and one trough.
**Key Points and Behavior:**
1. **Intercepts:**
- The graph intersects the x-axis at \( x = -2 \), \( x = 0 \), and \( x = 3 \).
2. **Shape:**
- The graph starts from the negative side of the y-axis, rises to a local maximum, decreases to a local minimum, and rises again, typical of a cubic function.
Given these points, the polynomial could be represented by a function of the form \( f(x) = a(x + 2)(x)(x - 3) \).
To fully determine the function, the coefficient \( a \) would need to be found, often through additional points or specific conditions which are not provided in the question description.
In expanded form, with \( a = 1 \) (assuming leading coefficient adjustments aren't specified):
\[ f(x) = (x + 2)x(x - 3) \]
\[ = (x^2 + 2x)(x - 3) \]
\[ = x^3 - 3x^2 + 2x^2 - 6x \]
\[ = x^3 - x^2 - 6x \]
This polynomial should match the given graph's shape and intercepts when plotted.
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