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Characteristics of increasing graph

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Describe the following characteristics of the graph shown in Graph3.pdf:

1. Where is the function increasing, decreasing, or constant?

2. Are there any relative/absolute extrema? If so, where?

3. Is the graph smooth or choppy (piecewise)?

4. Are there any restrictions on the domain?

5. Are there any horizontal or vertical asymptotes?

6. How are the ends of the graph behaving?

7. Where are the x- and y-intercepts?

8. Write the equation.

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Solution Preview

1. It's virtually impossible to tell the exact coordinates of the points where the graph reaches its "highs" and "lows," but the function is increasing on two intervals, which are given below (and whose endpoints should be considered estimates):

x < -10

-3 < x < 3:5

The function is decreasing on two intervals, which are given below (and whose endpoints should be considered estimates):

-10 < x < -3

x > 3.5

The function is probably not constant anywhere, unless the relatively long "horizontal line" at approximately x = -3 is actually indicative of a piecewise function (and not simply an artifact of the resolution afforded by the computer
or graphing calculator that was used to generate the graph).

2. There are relative maxima at approximately x = -10 and approximately x = 3.5; the relative maximum at approximately x = -10 is both a relative maximum and an absolute maximum. There is a relative minimum at approximately x = -3. There are no absolute minima, provided that the graph goes to negative infinity as x goes to ...

Solution Summary

The various characteristics of the given graph are explained in detail (including provision of caveats in regard to what can/cannot definitely be inferred from it).

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