Rate of change - constant or accelerating

Some functions change at a constant rate as x gets larger. Some functions change faster and faster as x gets larger. Some functions change slower and slower as x gets larger. Here are some possible categories of functions and their rates of change as x gets larger -  - functions that increase and their rates of change increase - functions that increase and their rates of change decrease - functions that decrease and their rates of change increase - functions that decrease and their rates of change decrease Give an example of a function that fits in each category. Are there functions that fit in NONE of these categories? Example, or why not? Are there functions that fit in MORE THAN ONE of these categories? Example, or why not? Does every possible function fit into at least one of these categories? Why, or why not? What questions could / would you ask your students based on this applet?
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