Concept explainers
To analyze: the given polynomial function
Answer to Problem 96AYU
(a)
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
Consequently,
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 of fare -1, 0 and 1. The zero-1 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.
Because 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.
The two x -intercepts are - 1,0, 1
Near -1,
This is a line with slope-2
Near 0,
This is a line making equal angles with the positive parts of both x and y axes
Near 1,
This is also a line with slope -2
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 -2, 2 to help establish the scale on the y -axis. The graph of f is given in Figure 1(b).
(a)
Conclusion:
Therefore, the polynomial function
Chapter 4 Solutions
Precalculus
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