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Graphing Rational Functions

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Example (1) For the following function: determine the intervals of increase/decrease, the intervals of upward/downward concavity, the points of extremum and inflection, the intersections with the axes and the asymptotes and then graph it.

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Asymptotes

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Intersection with the Axes & Extremum See Appendix

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Intervals of increase/decrease

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Another Way to show extremum First Derivative Test

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Concavity

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Graph

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Example (2) For the following function: determine the intervals of increase/decrease, the intervals of upward/downward concavity, the points of extremum and inflection, the intersections with the axes and the asymptotes and then graph it.

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Asymptotes

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Intersection with the Axes & Extremum See Appendix

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Intervals of increase/decrease

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Another Way to show extremum First Derivative Test

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Concavity

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Graph

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Example (3) For the following function: determine the intervals of increase/decrease, the intervals of upward/downward concavity, the points of extremum and inflection, the intersections with the axes and the asymptotes and then graph it.

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Asymptotes

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Intersection with the Axes & Extremum See Appendix

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Intervals of increase/decrease

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Another Way to show extremum First Derivative Test

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Concavity

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Graph

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Appendix Differentiating & Simplifying

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Example(1)

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Example(2)

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Example(3)

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1 Example 3 Sketch the graph of the function Solution Observe that h is an odd function, and its graph is symmetric with respect to the origin. I. Intercepts.

1 Example 3 Sketch the graph of the function Solution Observe that h is an odd function, and its graph is symmetric with respect to the origin. I. Intercepts.

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