We know the roots of the function include +/-1 and +3. So we would expect to see the factors:

(x+1)(x-1)(x-3)

All of the functions have (x-3) or equivalent factors like (3-x) or (1-x/3).

We can eliminate two functions, though, that don't have (x+1)(x-1) or equivalent (x^2 - 1). These are a) (has (x^2+1) as a factor) and e) (has (x^2-1)^2 instead; would have 5 roots hence 5 turning points instead of the 3 pictured).

Next we check the direction of the function: Since we have a cubic increasing for -ve x, and decreasing for +ve x, the first term coefficient (for x^3) must be negative. This rules out b).

So c), d) and f) are possible functions.

Mar 29th, 2015

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