POLYNOMIAL
FUNCTIONS
Chapter 5
5.1 – EXPLORING THE
GRAPHS OF POLYNOMIAL
FUNCTIONS
5.2 – CHARACTERISTICS
OF THE EQUATIONS OF
POLYNOMIAL FUNCTIONS
Chapter 5
POLYNOMIAL FUNCTIONS
What’s a polynomial?
A polynomial is any algebraic expression that features
addition, subtraction, multiplication and division (but not
division by a variable). It has real number coefficients,
whole number exponents, and two variables x and y or f(x).
In general, a polynomial equation is:
P( x)  ax  bx
n
n 1
 cx
n2
 ...  k
Where n=0 or any positive integer (no fractional or
negative exponents).
Examples of Polynomials
Not Polynomials
y  2x  7
f ( x)   2 x  6x  8
f ( x)   2 x 2  6x  8
P( x)   10x  7 x  6x
5
f ( x)  90x  12
3
4
2
y 3 6
x
2
P( x )   2 x 2  6 x
1
3
VOCABULARY
Degree: The value of the highest exponent of a
polynomial function.
f (x )  4x 3  7x  5
Leading Coefficient: The coefficient of the term
with the greatest degree (exponent) in a
polynomial function.
Constant Term: The term in a polynomial
function with no variable.
POLYNOMIAL FUNCTIONS ARE NAMED ACCORDING TO
THEIR DEGREE AND THEIR DEGREE DETERMINES
THE SHAPE OF THE FUNCTION.
DEGREE
NAME
EXAMPLE
0
Constant
y 7
1
Linear
2
Quadratic
3
Cubic
y  5x  11
y  x 2  7x  6
y  7x 3  2x
EXAMPLE 1. DETERMINE THE DEGREE, THE
LEADING COEFFICIENT, AND THE CONSTANT
TERM OF EACH POLYNOMIAL FUNCTION.
DEGREE
f ( x)  x 2  4 x  5
g ( x)  2 x  4
h( x)  3
j ( x)  x 3
LEADING
COEFFICIENT
CONSTANT
TERM
EXAMPLE 2. WRITE A POLYNOMIAL FUNCTION
IN DESCENDING ORDER THAT SATISFIES THE
FOLLOWING CONDITIONS .
A. Degree 2, leading coef ficient of -3
B. Degree 2, leading coef ficient of 7, two terms
C. Degree 1 , leading coef ficient of 1
D. Degree 0
E. Degree 3, constant term -8
PAGE 287 # 1 , 4
Independent
practice
VOCABULARY CONTINUED…
Domain: the set of all possible x-values which will make the
function “work” and will output real y -values.
Range: The complete set of all possible resulting y -values of
the dependent variable.
End Behaviour: This is the description of the shape of the
graph, from left to right, on the coordinate plane. It is the
behaviour of the y -values as x becomes large in the positive or
negative direction. i.e. as x    ).
x approaches ±∞
Turning Point: Any point where the graph of a function changes
from increasing (y -values) to decreasing (y -values) or from
decreasing to increasing.
HOW GRAPHS WORK
End behaviour: The description of
the shape of the graph, from left to
right, on the coordinate plane.
A Cartesian grid (the x/yaxis) has four quadrants.
Example: the graph of f(x) = x + 1
begins in quadrant III and extends to
quadrant I.
DOMAIN AND RANGE
Domain is how much of the x-axis is spanned by the graph.
Range is how much of the y-axis is spanned by the graph.
TURNING POINTS
A turning point is any point where the graph changes from
increasing to decreasing, or from decreasing to increasing.
INVESTIGATING GRAPHS OF CONSTANT
FUNCTIONS – DEGREE 0
Function
Graph of Function
# of x-intercepts
# of y-intercepts
End Behaviour
Domain
Range
Turning Points
y 3
f ( x)   2
h( x)  0
SUMMARY
y  b constant function
Degree
# of x-intercepts
# of y-intercepts
Domain
Range
# of Turning Points
INVESTIGATING GRAPHS OF LINEAR
FUNCTIONS – DEGREE 1
Function
Graph of Function
# of x-intercepts
# of y-intercepts
End Behaviour
Domain
Range
Turning Points
y  3x  1
f ( x)   2 x  3
h( x ) 
1
x
2
SUMMARY
y  mx  b linear function
Degree
# of x-intercepts
# of y-intercepts
Domain
Range
# of Turning Points
What’s the relationship between slope of a line and its end
behaviour?
INVESTIGATING GRAPHS OF QUADRATIC
FUNCTIONS – DEGREE 2
Function
Graph of Function
# of x-intercepts
# of y-intercepts
End Behaviour
Domain
Range
Turning Points
y  x 2  2x  8
f ( x)   x 2  2 x  2
y  x 2  4x  4
SUMMARY
y  ax 2  bx  c quadratic function
Degree
# of x-intercepts
# of y-intercepts
Domain
# of Turning Points
 How does the sign of the leading coefficient help us to determine
the end behaviour of a quadratic?

If the leading coefficient is positive, the graph goes from
Quadrant 2 to Quadrant 1 (opens up).

If the leading coefficient is negative, the graph goes from Quadrant
3 to Quadrant 4 (opens down).
 How does the sign of the leading coefficient help us to determine
the range of a quadratic?

If the leading coefficient is positive, { y/y> minimum}

If the leading coefficient is negative, { y/y< maximum}
INVESTIGATING GRAPHS OF CUBIC
FUNCTIONS – DEGREE 3
Function
Graph of Function
# of x-intercepts
# of y-intercepts
End Behaviour
Domain
Range
Turning Points
y   x 3  x 2  6x
y  x 3  3x  2
y  x 3  3x 2  3x  1
SUMMARY
y  ax 3  bx 2  cx  d cubic function
Degree
# of x-intercepts
# of y-intercepts
Domain
Range
# of Turning Points
 How does the sign of the leading coef ficient help us to
determine the end behaviour of a cubic polynomial function?
POLYNOMIAL FUNCTIONS
Pull out a graphing calculator.
Equation
Name /
Degree
End
Max # of xBehaviour intercepts
# of yintercepts
f(x) = 4
Constant
Deg 0
Q2 – Q1
0
1
f(x) = x + 1
Linear
Deg 1
Q3 – Q1
1
1
f(x) = x2 + 3x – 4
Quadratic
Deg 2
Q2 – Q1
2
1
f(x) = 5x3 + 5x2 – 4x –
2
Cubic
Deg 3
Q3 – Q1
3
1
 How would a negative leading coef ficient change the above
graphs?
PG. 287-289
# 2, 3, 5-13
Independent
practice
LET’S RECAP!
Determine the following characteristics of each function using its equation.
Number of possible x-intercepts
Domain
y-intercept
Range
End behaviour
Number of possible turning points
a) f(x) = 3x – 5
What’s the degree?
Linear equations (of degree 1) always have only one
x-intercept.
Where can I find the y-intercept?
The last number is always the y-intercept.
So, in this case, it’s –5.
Is the leading coefficient positive or negative?
A positive leading coefficient in
a linear equation means that
the graph starts in quadrant III
and goes to quadrant I.
There are 0 turning points in a linear equation.
STANDARD FORM
Linear Functions:
y = ax + b
Quadratic Functions:
slope
y-intercept
y = ax2 + bx + c
direction of opening
y-intercept
Cubic Functions:
y = ax3 + bx2 + cx + d
EXAMPLE
Determine the following characteristics of each function using its equation.
Number of possible x-intercepts
Domain
y-intercept
Range
End behaviour
Number of possible turning points
b) f(x) = –2x2 – 4x + 8
What’s the degree?
Quadratic functions (with degree 2), can either have 0, 1
or 2 x-intercepts. How many does this one have?
What’s the y-intercept?
The y-intercept is always the last number—in this case,
it’s 8.
The leading coefficient?
The leading coefficient is negative, so the graph
opens downwards. That means it extends from
quadrant III to quadrant IV.
The number of turning points for a
quadratic is 1.
EXAMPLE
Determine the following characteristics of each function using its equation.
Number of possible x-intercepts
Domain
y-intercept
Range
End behaviour
Number of possible turning points
c) f(x) = 2x3 + 10x2 – 2x – 10
What’s the degree?
Cubic functions (of degree 3) can have either 1, 2, or 3
x-intercepts.
The y-intercept is still the last number, in this case –10.
Leading coefficient?
A positive leading coefficient means that a cubic
function extends from quadrant III into quadrant I.
A cubic function could have 0 turning points or 2
turning points.
EXAMPLE
Sketch the graph of a possible polynomial function for each set of characteristics below.
What can you conclude about the equation of the function with these characteristics?
a)
b)
5.3 –
MODELLING DATA
WITH A LINE OF
BEST FIT
5.4 –
MODELLING DATA
WITH A CURVE OF
BEST FIT
Chapter 5
LINE OF BEST FIT
A line of best fit is a straight line that best approximates the
trend in a scatter plot.
A regression function is a line or curve of best fit, developed
through statistical analysis of data.
EXAMPLE 1
The one-hour record is the farthest distance travelled by bicycle in 1 h. The table below
shows the world-record distances and the dates they were accomplished.
a) Use technology to create a scatter plot to find the equation of the line of best fit.
b) Interpolate a possible world-record distance for the year 2006, to the nearest
hundredth of a kilometre.
c) Compare your estimate with the actual world-record distance of 85.99 km in 2006.
A) USING YOUR CALCULATOR TO CREATE
A SCATTER PLOT/LINE OF BEST FIT.
Entering data:
STAT
EDIT
Enter your x-values in under L1
and your y-values in under L2
A) USING YOUR CALCULATOR TO CREATE
A SCATTER PLOT/LINE OF BEST FIT.
Making a scatter plot:
2nd
Y=
ENTER
ZOOM
9
TO MAKE A LINE OF BEST FIT:
STAT
CALC
4
ENTER
Write down your equation in the form:
y = ax + b
.
y = 0.8584802632x – 1635.732803
GRAPH YOUR LINE OF BEST FIT:
B) INTERPOLATE A POSSIBLE WORLD -RECORD
DISTANCE FOR THE YEAR 2006, TO THE NEAREST
HUNDREDTH OF A KILOMETRE.
Is 2006 an x-value or a y-value?
y = 0.8584802632x – 1635.732803
c) Compare your estimate with the actual world -record distance
of 85.99 km in 2006.
EXAMPLE 2
Matt buys t-shirts for a company that prints art on t-shirts and then resells them. When
buying the t-shirts, the price Matt must pay is related to the size of the order. Five of
Matt’s past orders are listed in the table below.
a) Create a scatter plot and determine an equation
for the linear regression function.
b) What do the slope and y-intercept represent?
c) Use the linear regression to extrapolate the size
of the order necessary to achieve the price of
$1.50 per shirt.
x- and y-values
line of best fit
scatter plot
y=-.0065x+6.5
B) WHAT DO THE SLOPE AND Y-INTERCEPT
REPRESENT?
C) USE THE LINEAR REGRESSION TO EXTRAPOLATE
THE SIZE OF THE ORDER NECESSARY TO ACHIEVE
THE PRICE OF $1 .50 PER SHIRT .
MODELLING DATA WITH A QUADRATIC
CURVE OF BEST FIT
Audrey is interested in how speed plays a role in car
accidents. She knows that there is a relationship between the
speed of a car and the distance needed to stop. She would
like to write a summary of this data for the graduation class
website.
a) Plot the data on a scatter plot. Determine the equation of a
quadratic regression function that models the data.
y  0.008284799x 2  0.539874172x  10.449
B) USE YOUR EQUATION TO COMPARE THE STOPPING
DISTANCE AT 30 KM/H WITH THE STOPPING
DISTANCE AT 50 KM/H, LENGTH OF A CAR.
y  0.008284799x 2  0.539874172x  10.449
C) DETERMINE THE MAXIMUM SPEED THAT A CAR
SHOULD BE TRAVELLING IN ORDER TO STOP
WITHIN 4 M, THE AVERAGE LENGTH OF A CAR.
MODELLING DATA WITH A CUBIC CURVE
OF BEST FIT
The following table shows the average retail price of gasoline,
per litre, for a selection of years in a 30 -year period beginning
in 1979.
a) U S E T E C H NOLOGY TO G R A P H T H E DATA A S A S CATTE R
P LOT. W H AT P OLY N OM IAL F U N C TION C OU L D B E U S E TO
M OD E L T H E DATA? E X P L AIN .
B) DE T ERMI NE T H E C UBI C RE G RE SSI ON E QUAT ION T H AT
M OD E LS T H E DATA . U S E YOU R E QUATI ON TO E S T I MATE T H E
AV E R AG E P RI C E OF G A S I N 1 9 8 4 A N D 1 9 8 5 .
P  0.0123n 3  0.4645n 2  6.295n  23.452
c) Estimate the year in which the average price of gas
was 56.0¢/L.
PG. 301-306, #3, 4, 6, 7,
8, 11, 14
PG. 313-316, #1, 2, 3, 5,
7, 8, 9
Independent
practice