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Concept explainers
To analyze: the givenpolynomial function
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Answer to Problem 98AYU
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 3. The graph of f behaves like
Step2: find the x -and y -intercepts of the graph of the function
The y -intercept is
To find the x -intercepts, solve
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 -4, 0,2.
The root-4is a zero of multiplicity 1, so the graph off crosses the x -axis at
The root 0 is a zero of multiplicity1. So the graph of f crosses thex -axis at
The root 2 is a zero of multiplicity 1. So the graph of f crosses the x -axis at
Step 4: Determine the maximum number of turning points on the graph of the function.
While the polynomial function is of degree 3 (step1), the graph of the function will have at most
Step5: Determine the behavior of the graph of f near each x -intercept.
Thex -intercepts are - 4,0, and 2.
So, the behavior of the function near -4 is
This is a straight line with a positive slope 24.
Near
This is a straight line with a negative slope -8.
Near
This is a straight line with a positive slope 16.
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. Evaluate f at -2.5, l 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
Glencoe Math Accelerated, Student Edition
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