Tom K. answered • 07/05/20
Knowledgeable and Friendly Math and Statistics Tutor
The domain is all reals and, since this polynomial has an odd degree, (three factors with degree 1 means the product has degree 3), the range is all reals.
The polynomial's leading coefficient, 6, is positive and the degree is odd, so y goes to +∞ as x goes to +∞ and -∞ as x goes to -∞
x - 1 = 0, so x = 1; 2x - 3 = 0, or 2x = 3, or x = 3/2; 3x - 1 = 0, or 3x = 1, or x = 1/3
The roots are 1, 3/2, and 1/3
The polynomial will turn between roots, so there are two turning points. All roots are single roots (odd number), there are no reversals at the roots Thus, there are two reversals.
Given where the roots are, I started off plotting from -1 to 3, and found that a plot from 0 to 2 looks better.
In Excel, it is easy to increment x by .01. A graphing calculator will work. Make sure that you have the polynomial having a value of 0 at 1/3, 1, and 3/2. Calculate at 0, 1/2, and 2, and your polynomial can already look good. With calculus, you can find the exact relative maximum and relative minimum (turning points). Without calculus, you can use your graphing calculator to zoom in to the relative maximum and minimum. Just for grins, near the maximum and minimum, I used Excel's goal seek and got a relative maximum near .6065 and relative minimum near 1.2824.
(If this were later in your Calculus class, you could calculate the location of the quadratics exactly; As the original equation is a cubic, the derivative (slope) formula will be a quadratic, so you can use the quadratic formula to calculate the exact values.
