FM 30(IB)
Graphing Polynomial Functions – End Behaviour (Day 1)
For the following functions:
 Identify the coefficient and the degree of the leading term from the given equation in simplified form.
 Use your graphing calculator to display the graph of each function.
 Identify the y-intercept and the number of x-intercepts.
 Indicate the quadrants in which each graph begins and ends.
Equation of the
function
1.
y  3x  2
2.
y  x  2
3.
y  x 2  5x  6
y  x  3x  2
4.
y  x 2  x  2
y  x  1x  2
5.
6.
7.
y  x 2  4x  4
y  x  2
2
y  x 3  4x
y  xx  2x  2
y  x3  x2  x 1
y   x  1  x  1
2
y  x 3  6 x 2  12 x  8
8
y  x  2
3
y   x 3  2 x 2  5x  6
9.
y  x  2x  1x  3
Leading term
coeffic
ient
Rough Sketch
degree
yintercept
Number
of xintercepts
Quadrant in
which graph:
begins
ends
y  2 x 3  14 x 2  30 x  18
10.
11.
12.
13.
14.
15.
y  2 x  1 x  3
2
y  x 3  2x 2  4x  8


y   x 2  4  x  2
y  x 4  5x 2  4
y  x  1x  1x  2x  2
y  x 4  4x 2  4

y  x2  2

2
y  3x 4  11x 3  x 2  19 x  6
y  x  2x  33x  1x  1
y  x 4  x 3  3x 2  5 x  2
y   x  2  x  1
3
y   x 4  2 x 3  3x 2  4 x  4
16.
y  x  2 x  1
2
5
4
3
2
2
y  x  2 x  7 x  8 x  12 x
17.
18.
19.
y  xx  1x  2x  2x  3
y  3x 5  30 x 4  120 x 3  240 x 2  240 x  96
y  3x  2
5
y   x 5  x 4  3x 3  3x 2  4 x  4


y  x  1x  2x  2 x 2  1
y   x 6  x 4  10 x 3  8
20.


y  x  2x  2 x 2  1 x 2  2

2
Observations:
1. Look at the leading term for each function. Which functions have an odd degree and a positive leading
coefficient?
2. Based on your observations in question 1, the graph of a polynomial function with an odd degree and a
positive leading coefficient will begin in quadrant ____ and end in quadrant ____.
3. Which functions have an odd degree and a negative leading coefficient?
4. Based on your observations in question 3, the graph of a polynomial function with an odd degree and a
negative leading coefficient will begin in quadrant ____ and end in quadrant ____.
5. Which functions have an even degree and a positive leading coefficient?
6. Based on your observations in question 5, the graph of a polynomial function with an even degree and a
positive leading coefficient will begin in quadrant ____ and end in quadrant ____.
7. Which functions have an even degree and a negative leading coefficient?
8. Based on your observations in question 7, the graph of a polynomial function with an even degree and a
negative leading coefficient will begin in quadrant ____ and end in quadrant ____.
9. What is the domain of all of these functions?
A relationship exists between the number of possible x-intercepts and the degree of the polynomial function.
 Odd degree: Minimum of one x-intercept; maximum number of x-intercepts is indicated by the degree
of the polynomial function
 Even degree: Minimum of zero x-intercepts; maximum number of x-intercepts is indicated by the
degree of the polynomial function
Example: A polynomial function of degree of 6 may have no x-intercepts or as many as 6 x-intercepts.
A polynomial function of degree of 5 may have one x-intercept or as many as 5 x-intercepts.
3
Summary of Characteristics with a Positive Leading Coefficient:
Degree
Name
Even/Odd Number of
Degree x-intercepts
End Behaviour
Domain
0
1
2
3
4
5


The degree of a polynomial function is the greatest exponent of the variable x that exists in the
equation of the function.
The coefficient of this greatest power of x is the leading coefficient. The text book calls this term an .
The constant term represents the y  intercept of the function. The text book calls this term a0 .

In our course, the coefficients are restricted to being integral values, that is, they are integers.

 See FM30 text book page 382, graphs of polynomial function up to degree of 3. Domain, range, all
kinds of golden information!
Hmwk: Assignment 3 #2 – 4
FM30 text p. 383 #1 - 3
Graphing Polynomial Functions – End Behaviour (Day 2)
Ex.1a) Describe the end behaviour of the graph of the function f  x    x3  3x2  2x  1 . State the possible
number of x-intercepts, the y-intercept, and whether the graph has a maximum or minimum value.
b) Which of the following is the graph of the function f  x    x3  3x2  2 x  1 ?
4
2.
Given the polynomial y  2(x  1)2(x  2)(x  3)2, determine the following without graphing.
a) Describe the end behaviour of the graph of the function.
b) Identify the type of function. Determine the possible number of x-intercepts for this type of
function.
c)
How many x-intercepts does this function have? What are they?
d) Determine the y-intercept of the function.
y
40
35
30
25
20
15
10
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
Hmwk: Assignment 3 #1, 5 - 10
5