How do I solve and graph this problem?

Question

Graph each of the following polynomial functions. State the reasons for the way in which you draw your graphs.
f(x) = x^2(x − 1)(2x + 3)

Answer 1

First, find the zeros and y-intercept.

 

To find the zeros, set f(x)=0 and solve for x.  You will get

 

x = -2/3       ,      x = 0       ,       x = 1

 

x-intercepts = (-2/3, 0)  , (0,0)  , and (1, 0)

y-intercept = (0, 0)

 

 

In the interval (-∞, -2/3), f(x) is above the x-axis and decreasing.

In the interval (-2/3, 0), f(x) has a local maximum below the x-axis.

In the interval (0, 1), f(x) has a local minimum below the x-axis, with a local maximum at x=0.

In the interval (1, ∞), f(x) increases.

 

In the end, you should get a wavy W-shape curve.

Answer 2

Because the factor x is squared, the graph will touch, or 'bounce off' the x-axis at x=0.

The graph will cross the x-axis at x=1 and at x=-3/2. At the far right, the graph will be positive & rising.

At the far left, the graph will bepositive and rising (because degree of this polynomial is 4 (even) and leading coefficient is positive. The graph will be BELOW the x-axis on the intervals (-3/2,0) and (0,1).