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Concept explainers
To analyze: the given polynomial function
![Check Mark](/static/check-mark.png)
Answer to Problem 99AYU
Figure 1(a):
Figure 1(b):
Explanation of Solution
Given:
Calculation:
Let us consider the function
This polynomial function
Step1: Determine the end behavior of the graph of the function
Expand the polynomial to write it in the form
The polynomial function f is of degree 4. The graph of f behaves like
Step2: find the x -and y -intercepts of the graph of the function
Substitute
The y -intercept is
To find the x -intercepts, solve
If and only if any of the factors is zero.
So,
Step3: Determine the zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the x -axis at each x -intercept.
The zeros or roots of fare obtained from
That is
The root-6is a zero of multiplicity 1, so the graph off crosses the x -axis at
The root -2 is a zero of multiplicity1. So the graph of f crosses thex -axis at 2nd time.
The root 0 is a zero of multiplicity 1 and the curve crosses the x -axis for the 3rd time.
The zero 2 is a zero of multiplicity 1 and so, the curve crosses x -axis for the 4th time.
Therefore there are 3 turning points to the curve
Step 4: Determine the maximum number of turning points on the graph of the function.
While the polynomial function
Step5: Determine the behavior of the graph of f near each x -intercept.
Thex -intercepts with the function
So, the behavior of the function near -6 is
This is a straight line with a negative slope -384.
The behavior of f at -2 is:
This is a straight line with a positive slope 64.
The behavior of f at 0 is:
This is a straight line with a negative slope -48.
The behavior of f at 2 is:
This is a straight line with a positive slope 128.
Step 6: Put all the information from Steps 1 through 5 together to obtain the graph of f Figure 1(a) illustrates the information obtained from Steps 1 through 5. Evaluatef at -4.6, -1, 1.2 to help establish the scale on the y -axis.
The graph of f is given in Figure 1(b).
Figure 1(a):
Figure 1(b):
Conclusion:
Therefore, the polynomial function
Chapter 4 Solutions
Precalculus
Additional Math Textbook Solutions
University Calculus: Early Transcendentals (3rd Edition)
Thomas' Calculus: Early Transcendentals (14th Edition)
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
Calculus, Single Variable: Early Transcendentals (3rd Edition)
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- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning
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