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Polynomial Expression and Functions
Lesson #1: Characteristics of Polynomial Functions - Part 1
A POLYNOMIAL FUNCTION is an algebraic expression A relation is a FUNCTION iff for every x there
where all exponents of the variables are whole numbers and no is only 1 y value. A function must pass the VLT.
variable is contained in the denominator.
There are 3 types of functions we will study this semester:
Polynomial Exponential Periodic
A POLYNOMIAL FUNCTION can be written more generally as
where n is a whole number and a is a real number.
Notice that the polynomial function is written in terms of decreasing order of powers of x.
A polynomial function can be written as components, or POWER FUNCTIONS, . A power function is
the simplest type of polynomial function. So, if is the polynomial function, then the power functions
are a ,c .
Recall: Given the polynomial function, ,
the coefficients are 3, - 4, and 1, or that are all real numbers.
the degree of the function, n, which is the exponent of the greatest power of x with a non-zero
coefficient, is 2.
the leading coefficient is , or 3.
the constant term is , or 1.
Quadrants Read all graphs from left to right.
x-int occur where the relation crosses the x-
axis.
y-int occur where the relation crosses the y-
axis.
Domain is the set of x values that are
defined for the function.
Range is the set of y values that are defined
for the function.
Key Features of Standard Power Functions
Degree 1 Degree 2 Degree 3 Degree 4 Degree 5 Degree 6
ODD degree EVEN degree ODD degree EVEN degree ODD degree EVEN degree
Linear Function Quadratic Cubic Function Quartic Function Quintic Function 6th Degree
Function Function
x-int = 0 x-int = 0 x-int = 0 x-int = 0 x-int = 0 x-int = 0
y-int = 0 y-int = 0 y-int = 0 y-int = 0 y-int = 0 y-int = 0
The END BEHAVIOUR of is the behavior of the y-values as x increases or and as x decreases or
. NOTE: Infinity ( is not a number. It is a concept.
As , As , As , As , As , As ,
. . . . . .
As , As , As , As , As , As ,
. . . . . .
Starts in quadrant Starts in quadrant Starts in quadrant Starts in quadrant Starts in quadrant Starts in quadrant
3 and ends in 2 and ends in 3 and ends in 2 and ends in 3 and ends in 2 and ends in
quadrant 1. quadrant 1. quadrant 1. quadrant 1. quadrant 1. quadrant 1.
A graph has LINE SYMMETRY iff A graph has POINT SYMMETRY about
divides the graph into two parts that are reflections iff each part of the graph on one side of can be
rotated
of each other. to coincide with part of the graph on the other side
of
Point symmetry Line symmetry Point symmetry Line symmetry Point symmetry Line symmetry
about (0,0) along about (0,0) along about (0,0) along
Example 1: State the type of function each of the following equations represents.
a) b) c)
Polynomial Exponential Periodic
Example 2: Complete the table.
Graph
Equation
Degree 3 4 5
Type of Function Cubic Quartic Quintic
Domain
Range
As , . As , . As , .
End Behaviour As , . As , . As , .
Starts/Ends
Point Symmetry at Line Symmetry along Point Symmetry at
Line/Point
Symmetry
Polynomial Concept Attainment Activity
Compare and contrast the examples and non-examples of polynomial functions below.
Through reasoning, identify 3 attributes of every polynomial function that distinguish them from non-polynomial
functions:
a. _______________________________________
b. _______________________________________
c. _______________________________________
Examples Non Examples
yx y x
1
y 2x 1 f x 3x 2 x
2
y x x 6
5
y x2 x 2 y 2 16
y x 2 1 h x
2 3
x
f x x2 x y sin
1
y 0.2 4 x 3 x 3 y
x2
y x 3 2 x 2 x 11 y 2x
Polynomial Concept Attainment Activity (continued)
Examples Non Examples
x 1
y4 y
x x 1
2
1 2
h x x4 x 3
2
y 4 x 0 4
y x x 2 4 x 2
Lesson #2: Characteristics of Power Functions - Part 2
Even degree power functions have line symmetry along the y-axis or .
Odd degree power functions have point symmetry at the origin or .
Example 1: Complete the table.
(-1.7, 4.9) (2, 16)
Graph
(1.7,- 4.9) (-2, -128)
Equation
Leading -1 +1 +1
Coefficient
Constant 0 0 0 0
Degree 3 6 7 4
Type of Cubic 6 th Degree 7 th Degree Quartic
Function
As , As , As , As ,
End Behaviour
As , As , As ,
As ,
Starts/Ends
Line/Point Point Symmetry at Line Symmetry along Point Symmetry at Line Symmetry along
Symmetry
An interval can be expressed as an inequality or bracket interval.
-4 0 5 -7 0 4 -4 0 6
Domain
Examples
Range
Examples
Summary
Given an EVEN DEGREE power function with a POSITIVE LEADING
COEFFICIENT, the function will start in quadrant 2 and end in quadrant 1.
This function will have line symmetry along .
Given an EVEN DEGREE power function with a NEGATIVE LEADING
COEFFICIENT, the function will start in quadrant 3 and end in quadrant 4.
This function will have line symmetry along .
Given an ODD DEGREE power function with a POSITIVE LEADING
COEFFICIENT,
the function will start in quadrant 3 and end in quadrant 1.
This function will have point symmetry at .
Given an ODD DEGREE power function with a NEGATIVE LEADING
COEFFICIENT, the function will start in quadrant 2 and end in quadrant 4.
This function will have point symmetry at .
Lesson #3: Characteristics of Polynomial Functions - Part 1
This is the graph of the function . The LOCAL MAXIMUM is a “PEAK” and
it occurs where the slope of the function is
The (ABSOLUTE/GLOBAL) MAXIMUM is zero. The local max is approximately 8 and it
the largest value of y for the function which is defined or occurs when x is approximately -1.8.
there is no value.
The (ABSOLUTE/GLOBAL) MINIMUM is the The LOCAL MINIMUM is a “VALLEY”
smallest value of y for the function which is defined or there is and it occurs where the slope of the function is
no value. zero. The local min is approximately – 4 and it
occurs when x is approximately 1.
Example 1: Given the following graph of f(x), answer the following questions:
1.
e) The local max is _____________________ and occurs at x = ________.
f) The local min is _____________________ and occurs at x = ________.
g) The absolute max is _______________ and it occurs when x = ________.
h) The absolute min is _______________ and it occurs when x = ________.
2. a) The local max is _____________________ and occurs at x = ________.
b) The local min is _____________________ and occurs at x = ________.
c) The absolute max is _______________ and it occurs when x = ________.
Students are to complete on page 15 – 17 Investigate #1 during this period.
d) The absolute min is _______________ and it occurs when x = ________.
Investigate #1 What are the key features of the graphs of polynomial functions?
A: Polynomial Functions of Odd Degree
Group A Group B
Leading
Coefficient
End Behaviour
# of
Minimum Points
# of
Maximum Points
# of Local
Minimum Points
# of Local
Maximum Points
Similar to y = x
or y = - x?
Leading
Coefficient
End Behaviour
# of
Minimum Points
# of
Maximum Points
# of Local
Minimum Points
# of Local
Maximum Points
Similar to y = x
or y = - x?
Leading
Coefficient
End Behaviour
# of
Minimum Points
# of
Maximum Points
# of Local
Minimum Points
# of Local
Maximum Points
Similar to y = x
or y = - x?
2.
Leading
Coefficient
End Behaviour
# of
Minimum
Points
# of
Maximum
Points
# of Local
Minimum
Points
# of Local
Maximum
Points
Similar to y = x
or y = - x?
3. a) What are the similarities and differences between the graphs of linear, cubic, and quintic functions?
b) What are the minimum and the maximum numbers of x-intercepts of graphs of cubic polynomial functions?
c) Describe the relationship between the number of minimum and maximum points, the number of local minimum and
local maximum points, and the degree of a polynomial function.
d) What is the relationship between the sign of the leading coefficient and the end behavior of graphs of polynomial
functions with odd degree?
e) Do you think the results in part d) are true for all polynomial functions with odd degree? Justify your answer.
B: Polynomial Functions of Even Degree
Group A Group B
Leading
Coefficient
End Behaviour
# of
Minimum Points
# of
Maximum Points
# of Local
Minimum Points
# of Local
Maximum Points
Similar to y =
or y = - ?
Leading
Coefficient
End Behaviour
# of
Minimum Points
# of
Maximum Points
# of Local
Minimum Points
# of Local
Maximum Points
Similar to y =
or y = - ?
Leading
Coefficient
End Behaviour
# of
Minimum Points
# of
Maximum Points
# of Local
Minimum Points
# of Local
Maximum Points
Similar to y =
or y = - ?
2. a) What are the similarities and differences between the graphs of linear, cubic, and quintic functions?
b) What are the minimum and the maximum numbers of x-intercepts of graphs of cubic polynomial functions?
c) Describe the relationship between the number of minimum and maximum points, the number of local minimum and
local maximum points, and the degree of a polynomial function.
d) What is the relationship between the sign of the leading coefficient and the end behavior of graphs of polynomial
functions with an even degree?
e) Do you think the results in part d) are true for all polynomial functions with an even degree? Justify your answer.
Summary:
A polynomial function with n degree could have n – 1 local minimum and maximum points where .
Odd Degree
Positive Leading Coefficient Negative Leading Coefficient
Starts in Quad 3 and ends in Quad 1 Starts in Quad 2 and ends in Quad 4
At least 1 x-intercept, up to n x-intercepts
Maximum Point and
Minimum Point
Or no Maximum or Minimum Point
May have Point Symmetry
Even Degree
Positive Leading Coefficient Negative Leading Coefficient
Starts in Quad 2 and ends in Quad 1 Starts in Quad 3 and ends in Quad 4
At least 1 Min Point At least 1 Max Point
Zero to n x-intercepts
, range is dependent on max or min value
May have Line Symmetry
Lesson #4: Characteristics of Polynomial Functions - Part 2
Students are to complete Investigate 2 on page 17 – 18. Students will need a graphing calculator. Students are
responsible for making notes on the punching sequence for the TI83, TI83+ and/or TI84+ in order to create
differences and to regress.
Example 1: Use finite differences to determine:
a) the degree
b) the sign of the leading
coefficient
c) the value of the leading coefficient
Example 2: A medical researcher establishes that a patient’s reaction time, r, in minutes, to a dose of a particular drug is
, where d is the amount of the drub, in millimeters, that is absorbed into the patient’s blood.
a) What type of function is ?
b) Using a graphing calculator, determine the value of the constant differences.
c) Describe the end behavior of this function if no restrictions are considered.
d) State the restrictions for this situation.
Numerical Properties of Polynomial Functions
1. Consider the function y = x x y First Differences
a) What type of function is it?
b) Complete the table of values. –3
c) Calculate the first differences.
–2
d) In this case, the first differences
were positive. How would the
graph differ if the first differences –1
were negative?
0
1
2
3
2. Consider the function y x 2 Differences
x y
a) What type of function is it? First Second
b) Complete the table of values. –3
c) Calculate the first and second
differences. –2
–1
0
1
2
3
Numerical Properties of Polynomial Functions (continued)
3. Consider the function y x 3 Differences
x y
a) What type of function is it? First Second Third
b) Complete the table of values.
–3
c) Calculate the first, second,
and third differences.
–2
–1
0
1
2
3
4. Consider the Differences
function y x 4 x y
First Second Third Fourth
a) What type of
function is it? –3
b) Complete the –2
table of values.
c) Calculate the
–1
first, second,
third, and fourth
differences. 0
1
2
3
5. a) Summarize the patterns you observe in Questions 1–4.
b) Hypothesize as to whether or not your patterns hold when values for the b, c, d, and k parameters are
not equal to 0 in y ax k , y ax 2 bc k , y ax 3 bx 2 cx k , and y ax 4 bx 3 cx 2 dx k .
c) Test your hypothesis on at least 6 different examples. Explain your findings.
