Graph of polynomial function, xy-plane. Horizontal axis, scale -20 to 20, by 10’s. Vertical axis, scale -8,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 3, passes through about (-20 comma -8,000), ascending in a smooth curve flattened near and through the origin, flattened up into Quadrant 1 and ascends in a smooth curve to the upper right through about (20 comma 8,000).
Graph of polynomial function, xy-plane. Horizontal axis, scale -20 to 20, by 10’s. Vertical axis, scale -8,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 2, passes through about (-10 comma 8,000), descending in a smooth curve flattened near and through the origin, flattened up into Quadrant 1 and ascends in a smooth curve to the upper right through about (10 comma 8,000).
8.2
Activity
In the column for your assigned polynomial, evaluate for the different values of . Discuss what you notice with your group.
-1000
-100
-10
-1
1
10
100
1000
Sketch what you think the end behavior of your polynomial looks like, then check your work using graphing technology.
8.3
Activity
Consider the polynomial .
Identify the degree of the polynomial.
Which of the 6 terms, , , , , , or , is greatest when:
Describe the end behavior of the polynomial.
Student Lesson Summary
The value of the leading term determines the end behavior of the function, that is, how the outputs of the function change as we look at input values farther and farther from 0.
Consider the polynomial . The leading term, , almost seems smaller than the other three terms, and for certain values of , this is even true. But, for values of far away from 0, the leading term will always have the greatest value. In the case of , as gets larger and larger in the positive and negative directions, the output of the function gets larger and larger in the positive direction.
-500
62,500,000,000
3,750,000,000
-5,000,000
1,000
66,245,001,000
-100
100,000,000
30,000,000
-200,000
1,000
129,801,000
-10
10,000
30,000
-2,000
1,000
39,000
0
0
0
0
1,000
1000
10
10,000
-30,000
-2,000
1,000
-21,000
100
100,000,000
-30,000,000
-200,000
1,000
69,801,000
500
62,500,000,000
-3,750,000,000
-5,000,000
1,000
58,745,001,000
If we graph , and and zoom out, we see the following:
Graph of polynomial function y = x squared, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -2,000 to 4,000, by 1,000’s. Polynomial graph comes from Quadrant 2, passes down through (-50 comma 2,500), descending in a smooth curve through the origin, up into Quadrant 1 ascending in a smooth curve to the upper right through (50 comma 2,500).
Graph of polynomial function y = x cubed, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -8,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 3, passes up through (-20 comma 8,000), ascending in a smooth curve through the origin, up into Quadrant 1 ascending in a smooth curve to the upper right through (20 comma 8,000).
Graph of polynomial function, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -2,000 to 8,000, by 2,000’s. Polynomial graph comes from Quadrant 3, passes through (-10 comma 10,000), descending in a smooth curve through the origin, up into Quadrant 1 and ascends in a smooth curve to the upper right through (10 comma 10,000).
For both and , large positive values of or large negative values of each result in large positive values of .
But for , large positive values of result in large positive values of , while large negative values of result in large negative values of .
How the outputs of a function change as we look at input values further and further from 0.
This function shows different end behavior in the positive and negative directions. In the positive direction the values get larger and larger. In the negative direction the values get closer and closer to -3.
