Use graph sketching techniques to graph:  y= (x-2)(x+3)(x+5)   Use graph sketching techniques to graph: y= (x-2)(x+3)(x+5)

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lfryerda eNotes educator| Certified Educator

To graph polynomials we know that there are no asymptotes (since polynomials are very well-behaved) so we just need to use the remaining graphing techniques.

(1) Intercepts.  The function is in factored form, so the x-intercepts are found when y=0, which is at x=2, x=-3 and x=-5.  The y-intercept is at x=0 which is at `y=(-2)(3)(5)=-30`.

(2) Extrema. Expanding out the polynomial gives `y=x^3+6x^2-x-30`.  The derivative is now easy to find using the power rule.  `y'=3x^2+12x-1`.  Using the quadratic formula, the zeros are at `x=-2+-sqrt{39}/3 approx 0.08, -4.08`.  The second derivative is `y''=6x+12=6(x+2)`.  This means from the second derivative test that the root at `x approx 0.08` is a local minimum and the root at `x approx -4.08` is a local maximum.

(3) Inflection Points. The inflection point is found from the zero of the second derivative, which is at x=-2.  Since this is a simple root, the second derivative changes sign on either side of the inflection point, so it goes from concave down to concave up.

(4) Infinity. Since this is a polynomial, we can tell that as x goes to the right, the function goes up, and as x goes to the left, the function goes down.  This is the behaviour of the leading term `x^3`.

Putting this together we get the graph: