



 
Firstly, it's not clear whether you want this for one value of a or for a set of different values. A good representative set should include positive, negative, and zero values of a. It should also give values of e^ax that will neither explode or disappear into the xaxis within too much of the range where the polynomial part, p(x) = (2x^{2}  3x), is interesting. That factors as (2x  3)x so the range should include the roots at 0 and 2/3. One decent choice would be 2 < x < 2, a = 1,0,1. Now, sketch the parabola y = p(x) = (2x  3)x which is upward curving and crosses the x axis at 0 and 2/3. At 2 it takes the value 14; at 2, 2. Now pick one value for a; the easy one is 0. e^{0x} = 1 for all x, so you have already sketched the a=0 case. Try again with a=1. Find a few values of e^{x} in the interval [2,2]; either use a calculator or estimate e ~ 3 for a rough sketch. Find or estimate e^{2} p(2), e^{1} p(1), e^{0} p(0)=0, e^{1/2} p(1/2), e^{2/3}p(2/3=0, e^{1}p(1), and e^{2}p(2), and joint them up with a smooth curve. If you are using calculus techniques differentiate (2x^{2}3x)e^{x} and set it equal to 0 to find the local maximum and local minimum. Note that exponential functions "overpower" polynomials so e^{x} p(x) > 0 as x> infinity [horizontal asymptote]. Sketch, trying for a clean smooth freehand curve. Repeat for a=1. Good Hunting!  


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