
Honors Math 3 Name:
Date:
Graphing Polynomials: End Behavior
Investigation
End behavior is the behavior of the graph of a function as x becomes infinitely large (

or

).
You will need: a graphing calculator. Sketch a graph of each function and discuss with your groups
 f ( x )  x 2 f ( x )  x 3 f ( x )  x 4 f ( x )  x 5
  
Does your conjecture change for the group of functions below? f ( x )
  x
2 f ( x )
  x
3 f ( x )
  x
4 f ( x )
  x
5
Does your conjecture change for the group of functions below? f ( x )

2 x
2 f ( x )

2 x
3 f ( x )

2 x
4 f ( x )

2 x
5
Summarize your conjecture: f ( x ) f ( x )

  x
6 f ( x )
 x
6
2 x
6


Formalize your conjecture about the end behavior of a function of the form f ( x )
 ax n
for each pair of conditions below. a.
when a

0 and n is even b.
when a

0 and n is even
  c.
when a

0 and n is odd d.
when a

0 and n is odd
 
If a polynomial function is written in standard form, coefficient is f ( x )
 a n x n  a n

1 x n

1 
...
 a
1 x
 a
0
, the leading a n
. That is, the leading coefficient is the coefficient of the term of greatest degree in the polynomial.






Problem Set
1. Sketch the end behavior of the following functions. It’s okay if you’re not sure about the middle of the graph. a. f ( x )

3 x
4 
7 x
3 
13 b. f ( x )
 
2 x
3 
3 x
2 
4 x

1
 
2. Describe the end behavior of the graphs of the following functions. Decide if f ( x )

/
 
as x

/
 
. You do NOT need to completely distribute the ones that are factored—all that matters is the term with the highest power of x . as x

, f ( x )
 as x

, f ( x )

? a. f ( x )
 
2 x
3 
6 x

11 b. f ( x )
 x
4 
5 x
3  x
2 
2 x

1 c. f ( x )

2( x

1)( x

3)( x

5)
  d. f ( x )
  x
2
( x

7)( x

3) e. f ( x )

3 x
2
( x

1)
3
( x

2)
2


3. The four functions below have each been graphed. Match the graphs with the functions without using your calculator. Use the x-intercepts and end behavior to guide you.
f ( x )

( x

3)( x

1)( x

5) g ( x )

0.5( x

5)( x

1)( x

4) h ( x )
  x ( x

3)( x

3) k ( x )
 
1( x

3)( x

1)( x

5)
30 30
A
20

B
20
-8 -6
C
-4
10
-2
0
0
-10
-20
-30
30
20
2 4 6 8 -8
D
-6 -4
10
-2
0
0
-10
-20
-30
30
20
2 4 6 8
-8 -6 -4
10
-2
0
0
-10
-20
2 4 6 8 -8 -6 -4
10
-2
0
-10
0
-20
2 4 6 8
-30 -30
4. Sketch a graph of each of the functions below. Plot the x - and y -intercepts and make sure that your end behavior is appropriate. You do not need to plot any additional points. The x-scale is one unit; the y-scale is ten units. a. f ( x )

2( x

5)( x

1)( x

4) b. f ( x )
 
0.5( x

6)( x

2)( x

5)( x

1)

50
-8 -6
30
-4
10
-2
-10
0
-30
-50
2 4 6

8
50
-8 -6
30
-4
10
-2
-10
0
-30
-50
2 4 6 8
3

 c. f ( x )
 
3 x ( x

2)( x

6) d. f ( x )

2 x ( x

3)( x

1)(2 x

7)
50 50
30
-8 -6 -4
10
-2
-10
0 2 4 6
 30
8 -8 -6 -4
10
-2
-10
0 2 4
-30 -30

-50 -50
On the next two, you will need to factor the quadratic terms to find all the zeros. e. f ( x )

( x

5)( x

1)( x 2 
3 x

2) f. f ( x )
 
3( x 2 
2 x

8)(2 x 2 
9 x

5)

50 50
6
30
-8 -6 -4
10
-2
-10
0
-30
2 4 6
30
8 -8 -6 -4
10
-2
-10
0
-30
2 4 6

-50 -50
You need to factor the next two fully before graphing. g. f ( x )
 x 3 
2 x 2 
15 x h. f ( x )
 
3 x 3 
27 x
50 50

30 30
-8 -6 -4
10
-2
-10
0
-30
2 4 6 8 -8 -6 -4
10
-2
-10
0
-30
-50 -50
2 4 6
8
8
8

6. Write a possible equation for each graph below. Use the form f ( x )
 a ( x

__)( x

__)..... You do not need to find the value of a , but the end behavior should enable you to tell whether it is positive or negative. a. b.

30 30
20 20
-8 -6 -4
10
-2
0
-10
0
-20
2 4 6 8 -8
-30 c. d.
60
-8 -6 -4
40
20
-2
0
0
-20
-40
-60
2 4 6 8 -8
-6
-6
-4
-4
10
-2
0
-10
0
-20
-30
30
20
10
-2
0
-10
0
-20
-30
2
2
4
4
6
6
8
8
5