Concept explainers
To analyze: the graph of given function
Answer to Problem 40AYU
Explanation of Solution
Given:
Calculation:
Here, we have to analyze the graph of the rational function
Let us consider it as
Too, so that it will be easy to graph it on the Cartesian plane.
This can be done simply in a few steps. Let us do it.
Step1: We will factorize the numerator and denominator and find the domain of the rational function. Therefore,
Step2: Now, we will write
Step3: We have to find the intercepts on the graph. The x intercept can be found by determining the real zeroes of the numerator of
No real solutions.
The y intercept can be found by determining the value of
Step4: Let us now test for the symmetry of the graph. A graph is symmetric about
From this, we conclude that there is symmetry with respect to with
Step5: Now, we will locate the vertical asymptotes. This is done by finding the zeroes of the denomination of the rational function in lowest terms. With R in the lowest terms we find that the graph of
Step6: Now let us locate any horizontal or oblique asymptote and determine the point if any which intersect the asymptotes at any point.
There is no horizontal asymptote of the graph. Let us calculate the oblique asymptotes. This can be found by dividing the numerator by denominator and the part except the remainder is the asymptote.
In
Therefore, the polynomial has oblique asymptote as
Step7: Now let us graph
Step8: Now let us use the results we have from step 1 to 7 and the make the final graph by hand. The graph for this can be shown as:
Conclusion:
Thus, the graph of
Chapter 4 Solutions
Precalculus
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