Learning Resources

Shapes of Polynomial Functions

Step A

• If there are three real and unequal roots, the graph will cross the x-axis at the value of each root.  The graph will begin in the third quadrant and finish in the first, providing a > 0. In which quadrant do you think the graph will begin and end if a < 0?
• If there are three equal roots, the graph will cross the x-axis at the value of the root. It will actually "flatten out" around this value. The graph will begin in the third quadrant and finish in the first. It is a horizontal translation of the graph y = x³. If, however, a < 0, then the graph is obtained from a reflection, as well as a horizontal translation, of the graph of y = x³. Thus , the graph will start in the second quadrant and end in the fourth.