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I'm confused by the following related to a course I'm doing in grade $12$ functions:

Consider polynomial functions in the form $f(x) = ax^n + x$. Use at least three examples to investigate the relationship between $a$, $n$, and the constant finite differences. Explain the steps you took in your investigation and any conclusions you found.

I realize that finite differences will indicate the degree of the polynomial, i.e $x^1$ (linear) will have first differences equal, $x^2$ (quadratic) will have second differences equal, $x^3$ (cubic) will have $3$rd differences equal, etc., but am unsure how to answer this question. Any advice is appreciated.

FishDrowned
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Dan Murray
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    I think you have the hardest part already. Looks like they just want you to explore what happens for different values of $a$ and $n$. I would try graphing several and just noticing their behavior (does the value of the function go upwards to $\infty$ or downards to $-\infty$ on the left/right sides, does $a$ change the shape or just scale and/or reflect the graph, etc). It's not a very precise question so I don't think there is a unique correct answer here. – pancini Nov 19 '23 at 05:20

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