Question

Based on the shape of the graph above, describe one or more functions you can think of to model the data. (Hint: Think of the graph in multiple parts.)

Answer 1

Answer:

  A polynomial together with the absolute value function can make a pretty good model (see the attachment)

Step-by-step explanation:

With a sufficient number of specified points, a polynomial can make a pretty good model of almost any smooth function. Here, the function's derivative is undefined at a couple of points, so there are some options for those. If the slopes match on either side of those zeros, then the absolute value function can be used to model the "reflection" at the x-axis. Otherwise, a piecewise description can be used.

The left portion of the curve looks a little like a sine wave, but a cubic or other polynomial can model that wave fairly well. The portion to the right of the maximum looks like a bouncing ball, so can be modeled by a piecewise quadratic function.

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Comment on the graph

The attached graph is of a 9th-degree polynomial model. Points were added until the model almost matched the curve. If additional points are defined, a higher-degree model may give a better fit. The "bounce" is modeled by the absolute value function.