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Is it possible to interpolate a polynomial that approximates Sin, and then be able to manipulate the polynomial or a sample of its points to make sections of the plotted curve progressively steeper?

Basically, I'd like to be able to make a plot showing, over a suitable domain, Sin and Abs[Sin[]] and everything in between.

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What does making sine steeper have to do with its absolute value? It sounds like you want to interpolate between the two, which might be done with f[t_, x_] := (1-t) Sin[x] + t Abs[Sin[x]]; for t between 0 and 1, you get a function of x between Sin[x] and Abs[Sin[x]]. –  Michael E2 Aug 6 '13 at 19:28
In general appearance, I want the first graph to look like a sine wave and the last graph to look like abs(sine). Abs(sine) is very "pointy." I want to be able to gradually progress to that "pointiness" from the sine curve. –  meihua Aug 7 '13 at 15:16
Do you insist on a polynomial approximation? See my nonpolynomial answer below. (This question seems to be more about mathematics and not about the features of Mathematica per se.) –  Michael E2 Aug 7 '13 at 15:28

1 Answer 1

Implementing my comment, a linear interpolation between sine and its absolute value:

 Plot[(1 - t) Sin[x] + t Abs[Sin[x]], {x, -2 Pi, 2 Pi}, PlotRange -> 1],
 {t, 0, 1}

enter image description here

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