Investigating End Behavior
Use your calculator to graph each polynomial and complete the information below each graph. You will
be using this information to look for and express regularity in repeated reasoning.
A. f ( x)  x  2
B. f ( x)  2( x  3)( x  3)
C. f ( x)   ( x  3)( x  3)
Zeroes:{-2}
Degree of polynomial:1
End behavior: down, up
Zeroes: {-3, 3}
Degree of polynomial: 2
End behavior: up, up
Zeroes:{-3, 3}
Degree of polynomial: 2
End behavior: down, down
D. f ( x)  ( x  2)( x  1)( x  4)( x  5) E. f ( x) 
Zeroes: {-2, 1, 4, -5}
Degree of polynomial: 4
End behavior: up, up
1
( x  2)( x  3)( x  1)
3
F. f ( x)   ( x  2)( x  1)( x  1)
Zeroes: {-2, 3, 1}
Degree of polynomial: 3
End behavior: down, up
Zeroes: {-2, 1, -1}
Degree of polynomial: 3
End behavior: up, down
G. f ( x)  x ( x  2)( x  1)( x  3)( x  4)
H. f ( x)   x ( x  2)( x  1)( x  3)( x  4)
Zeroes: {0, -2, 1, 3, -4}
Degree of polynomial: 5
End behavior: down, up
Zeroes: {0, -2, 1, 3, -4}
Degree of polynomial: 5
End behavior: up, down
1. Explain how you would predict the end behavior of a given polynomial of degree n.
Descriptions may vary.
Even Degree: Same end behavior
Odd Degree: Opposite end behavior
Positive Leading Coefficient: Right end behavior is up
Negative Leading Coefficient: Right end behavior is down
2. How do coefficients in front of the polynomial affect the end behavior?
Positive Leading Coefficient: Right end behavior is up
Negative Leading Coefficient: Right end behavior is down
Absolute value of the leading coefficient does not affect the end behavior (ex. B and E)
3. Using your prediction, sketch the following graphs without using a calculator.
I. f ( x)  ( x  4)( x  2)( x  3)( x  4) J. f ( x)   3 ( x  3)( x  1)
Zeroes: {-4, 2, -3, 4}
Degree of polynomial: 4
End behavior: up, up
L. f ( x)   ( x  2)( x  1)( x  4)
Zeroes: {-2, 1, 4}
Degree of polynomial: 3
End behavior: up, down
K. f ( x)  ( x  2)( x  1)( x  4)
Zeroes: {-3, 1}
Degree of polynomial: 2
End behavior: down, down
M. f ( x) 
2
( x  2)( x  3)( x  1)
3
Zeroes: {-2, 3, 1}
Degree of polynomial: 3
End behavior: down, up
Zeroes: {-2, 1, -4}
Degree of polynomial: 3
End behavior: down, up
N. f ( x)   x( x  2)( x  1)( x  1)
Zeroes: {0, -2, 1, -1}
Degree of polynomial: 4
End behavior: down, down
4. Compare your answers with others. Then make any necessary adjustments to your original conjectures below.
This activity was adapted from Jordan School District. Original available at http://secmathccss.wordpress.com/.