UNIT 1

DAY 1 - Polynomial Functions
x
x
n
A power function is a function of the form f x ax , where a z 0 , and n is a positive integer.
A polynomial function is a function whose equation is in the form of f x a n x n a n 1 x n 1 a n 2 x n 2 ... a 2 x 2 a1 x a0 ,
a ’s are coefficients and real numbers,
where
x is a variable,
n is a degree of a function (greatest power) and is a natural number,
a n is the leading coefficient,
a 0 is the constant term.
For example: f x 5 x 4 7 x 3 x 1 is a polynomial function. The coefficients must be real numbers and the
exponents must be whole numbers.
x
A polynomial function is defined explicitly using just one variable. In the example above 'x' is the independent variable
that defines the function. So, x2 + y2 = 25 is NOT a polynomial function because the function is not defined explicitly in
terms of x nor is it defined explicitly in terms of y.
x
The exponents on the variable must be whole numbers { 0, 1, 2, 3, . . . }. So
1
y
1
x 3 is not a polynomial function. Nor is y
x since this would be written y
x2
x
The domain of a polynomial function is ALWAYS the set of real numbers: {x R}. So radical functions like y
not polynomial functions.
x
The range of a polynomial function can:
x are
* be the set of real numbers, {y R}
* have only an upper bound {y R| y < 5}
* have only a lower bound {y R| y > 2}.
It CANNOT have both an upper bound and a lower bound. So, x2 + y2 = 25 is not a polynomial function because it is a circle
with both an upper and lower bound of 5.
x
Graphs of polynomial functions do not have asymptotes. So, reciprocal functions like y 1 are not polynomial functions
x
because they have asymptotes.
Degree of a Polynomial Function:
Polynomials are named according to their degree.
Degree 0
Horizontal Line
Degree 1
Linear Function
Degree 2
Quadratic Function
Degree 3
Cubic Function
Degree 4
Quartic Function
Degree 5
Quintic Function
y=a
y = ax + b
y = ax2 + bx + c
y = ax3 +bx2 +cx +d
y = ax4 + bx3 + cx2 +dx + e
y = ax5 +bx4 + cx3 + dx2 +ex +f
The End Behaviour of the graph is the behaviour of the y-values as x increases or approaches positive infinity x o f and as x
decreases or approaches negative infinity x o f .
that divides the graph into two parts such that each part is a
A graph has line symmetry if there is a vertical line x
(e.g. Quadratic function)
reflection of the other in the vertical line x
A graph has point symmetry about a point a, b if each part of a graph on one side of point a, b can be rotated 180 0 to
coincide with part of the graph on the other side of point a, b (e.g. Cubic function)
Polynomial Functions
y
,
,
n is even
Domain
End Behaviour
Polynomial Functions
End Behaviour
Extends from
Q3 to Q1
Extends from
Q2 to Q4
Extends from
Q2 to Q1
Extends from
Q3 to Q4
Function(s)
Reasons
Polynomial Functions
3x-4
Bracket
x > -2 & x
Y
N
Degree:
Lead. Coef:
Y
N
Degree:
Lead. Coef:
Y
N
Degree:
Lead. Coef:
Y
N
Degree:
Lead. Coef:
Polynomial Functions
y = 0.5x3 + 1
DAY 2 - Propertie
There are many characteristics that can be used to describe
polynomial functions. We will address SOME of them here.
1. Domain & Range: State domain & range of the functions:
g(x) = x4 - 3x3 - 6x2 + 8x +35
D:
R:
f(x) = x3 - 2x2 - 11x +12
D:
R:
Propertie
2. END BEHAVIOUR
Quadrants:
Words: As x approaches negative infinity, y approaches negative
infinity, and as x approaches positive infinity, y approaches
negative infinity.
Symbols:
Quadrants:
Words:
Symbols:
Propertie
2. END BEHAVIOUR cont'd
Quadrants:
Words:
Symbols:
Quadrants:
Words:
Symbols:
Propertie
3. ABSOLUTE MAXIMUM AND MINIMUM VALUES
An absolute maximum occurs at the point on the top of a "hill" that contains the
highest y-value on the graph.
An absolute minimum occurs at the point on the bottom of a "valley" in the graph
that contains the lowest y-value on the graph.
Minimum:
Minimum:
Minimum:
Maximum:
Maximum:
Maximum:
Propertie
4. LOCAL (Relative) MAXIMUM AND MINIMUM VALUES
A relative maximum occurs at the point on the top of a "hill". It is the greatest
value of a function in its neighbourhood.
A relative minimum occurs at the point on the bottom of a "valley". It is the least
value of a function in its neighbourhood.
Notice that there are other y-values on the graph that are greater than local
maximum or lesser than local minimum.
Minimum:
Minimum:
Local Minimum:
Local Minimum:
Maximum:
Maximum:
Local Maximum:
Local Maximum:
Propertie
5. INCREASING INTERVALS & DECREASING INTERVALS
Specify the intervals where the given function increases:
Specify the intervals where the given function decreases:
6. X-INTERCEPTS AND Y-INTERCEPTS
Using the graph of a function above state its x- and y-intercepts.
Propertie
COMPLETE THE FOLLOWING
INVESTIGATION OF PROPERTIES
OF POLYNOMIAL FUNCTIONS.
Investigation of the Properties of Polynomial Functions:
1. Sketch the functions using a graphing calculator and complete the chart:
End behaviour
Sign of leading
Function
Degree
xo f
coefficient
y
x3 2 x
y
x3 8x 2 4 x 1
y
2 x 5 7 x 4 3 x 3 18 x 2 5
y
5 x5 5 x 4 2 x3 4 x 2 3x
End behaviour
x o f
Conclusions about Odd Degree Polynomials:
If a polynomial function has an odd degree and its lead coefficient is positive, then the function
extends from the ______ quadrant to the ______ quadrant. Therefore: as
x o f, y o
x o f, y o
If a polynomial function has an odd degree and its lead coefficient is negative, then the function
extends from the ____ quadrant to the ______ quadrant. Therefore: as
x o f, y o
x o f, y o
2. Sketch the functions using a graphing calculator and complete the chart:
Function
y
3x 2 4 x 1
y
x 4 x3 x 2 3
y
2 x 6 9 x 4 11x 2 x 13
y
2 x 4 4 x 3 x 2 6 x 5
Degree
Sign of leading
coefficient
End behaviour
xo f
End behaviour
x o f
Conclusions about Even Degree Polynomials:
If a polynomial function has an even degree and its lead coefficient is positive, then the function
looks something like a parabola that opens __________________. The graph extends from the
______ quadrant to the ______ quadrant. Therefore: as
x o f, y o
x o f, y o
If a polynomial function has an even degree and its lead coefficient is negative, then the function
looks something like a parabola that opens __________________. The graph extends from the
______ quadrant to the ______ quadrant. Therefore:
x o f, y o
x o f, y o
3. Using a graphing calculator, fill-in the following charts and draw appropriate conclusions:
a) Quadratic Functions:
Function
y
x2
y
x2 1
y
3x 2 4 x 1
Degree
Number of zeroes/
x-intercepts/ roots
Number of Turning Points
(Max/Min)
Conclusions:
Quadratic functions have a degree of _____.
The maximum number of roots that a quadratic function can have is _____
The least number of roots that a quadratic function can have is _____
The maximum number of turning points (max/min) a quadratic function can have is _____
b) Cubic Functions:
Function
y
x3
y
x3 2 x 2 x 2
y
4 x 3 16 x 2 13 x 3
Degree
Number of zeroes/
x-intercepts/ roots
Number of Turning Points
(Max/Min)
Conclusions:
Cubic functions have a degree of _____.
The maximum number of roots that a cubic function can have is _____
The least number of roots that a cubic function can have is _____
The maximum number of turning points (max/min) a cubic function can have is _____
c) Quartic Functions:
Function
y
x4 5x2 4
y
x 4 3x3 x 2 3x 2
y
x4 5
y
x4
Degree
Number of zeroes/
x-intercepts/ roots
Number of Turning Points
(Max/Min)
Conclusions:
Quartic functions have a degree of _____.
The maximum number of roots that a quartic function can have is _____
The least number of roots that a quartic function can have is _____
The maximum number of turning points a quartic function can have is _____
d) Quintic Functions:
Function
y
x5 7
y
2 x 5 7 x 4 3 x 3 18 x 2 5
y
5 x5 5 x 4 2 x3 4 x 2 3x
Degree
Number of zeroes/
x-intercepts/ roots
Number of Turning Points
(Max/Min)
Conclusions:
Quintic functions have a degree of _____.
The maximum number of roots/x-intercepts that a quintic function can have is _____
The least number of roots/x-intercepts that a quintic function can have is _____
The maximum number of turning points a quintic function can have is _____
e) 6th Degree Functions:
Function
y
x6
y
2 x 6 12 x 4 18 x 2 x 5
y
x6 3
Degree
Number of zeroes/
x-intercepts/ roots
Number of Turning Points
(Max/Min)
Conclusions:
The maximum number of roots/x-intercepts that a 6th degree function can have is _____
The least number of roots/x-intercepts that a 6th degree function can have is _____
The maximum number of turning points a 6th degree function can have is _____
Overall Conclusions:
Number of Zeros:
The maximum number of zeros/x-intercepts that a polynomial function can have is the _____________
as its _________________________.
The minimum number of zeros/x-intercepts that an odd degree polynomial can have is _______.
However, an even degree polynomial function may have ______ zeros/x-ints.
Turning Points:
The maximum number of turning points that a polynomial function can have is
An even degree function must have at least ___________ turning point.
An odd degree function could have _________ turning points at all.
Complete the chart:
Type of
Polynomial
Degree
Maximum Number
of zeros /
x-intercepts / roots
Minimum Number
of Zeros/
x-intercepts/roots
Linear
Quadratic
Cubic
Quartic
Quintic
6
7
If n is even:
n
If n is odd:
Maximum Number
of Turning Points
Let’s See if you got it…
Take a look at the following relations and provide missing information:
Domain:
Domain:
Range:
Range:
Number of roots:
Number of roots:
Roots:
Roots:
End Behaviour:
End Behaviour:
Least Possible Degree:
Name:
Least Possible Degree:
Number of turning points:
Number of turning points:
Intervals of increase/decrease:
Intervals of increase:
Intervals of decrease:
Maxima/Minima:
Maxima/Minima:
Function?
Reasoning:
Function?
Reasoning:
Name:
4
2
-4
-3
-2
-1
1
2
3
-2
-4
-6
-8
-10
Domain:
Domain:
Range:
Range:
Number of roots:
Number of roots:
Roots:
Roots:
End Behaviour:
End Behaviour:
Least Possible Degree:
Name:
Least Possible Degree:
Number of turning points:
Number of turning points:
Intervals of increase:
Intervals of increase:
Intervals of decrease:
Intervals of decrease:
Maxima/Minima:
Maxima/Minima:
Function?
Reasoning:
Function?
Reasoning:
Name:
4
5
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Additional Worksheet
1. Determine the key features of the graphs of each polynomial function. Use these features to match each
function with its graph. Think, how are these features related to the degree of the function? How do they help you
to recognize a degree of polynomial?
a) fx x 3 2x
b) gx 3x 4 4x 3 4x 2 5x 5
c) hx 2x 5 7 x 4 3x 3 18x 2 5x 1
d) px 2x 6 12x 4 10x 2 x 10
1
2
* Number of x-intercepts:
* Number of x-intercepts:
* Circle one:
Maximum
Minimum
* Circle one:
None
* Number of Local Min's:
Local Max's:
* Number of Local Min's:
Maximum
Minimum
Local Max's:
* Number of turning points:
* Number of turning points:
* Least Possible Degree of polynomial:
* Least Possible Degree of polynomial:
3
4
* Number of x-intercepts:
* Number of x-intercepts:
* Circle one:
None
Maximum
* Number of Local Min's:
Minimum
None
Local Max's:
* Circle one:
Maximum
* Number of Local Min's:
Minimum
None
Local Max's:
* Number of the
turning
points: chart:
Complete
following
* Number of turning points:
* Least Possible Degree of polynomial:
* Least Possible Degree of polynomial:
2. Complete the following chart:
Equation
a)
fx x 4 2x 2 1
b)
gx x 3 3x 2 2x 5
c)
hx 1
2
d)
fx x 3 x
e)
gx 2x 6 3x 4
f)
hx x 5 3x
g)
fx x 2 3x 4
h)
gx 2x 7 3x 3 2x
i)
hx 3x 4 2x 3 3x 1
j)
fx x2 x
Degree
Even or
Odd
Degree?
Leading
Coefficient
x 10 31 x 4 x 2
3. Use finite differences to determine
a) the degree of the polynomial
function
b) the sign of the leading coefficient
c) the value of the leading coefficient
x
-3
-2
-1
0
1
2
3
y
140
37
8
5
4
5
32
End Behaviour
x o f
xof
Max # of
Turning
Points
# of
Turning
Points
DAY 3 - Equations and Graphs of Poly. Functions
Equations and Graphs of Poly. Functions
Consider the function f(x) = -2x(x - 3)(5x + 12)(x + 1).
i) the degree:
ii) the sign of leading coefficient:
iii) end behaviour
x -, y _____
x , y _____
iv) x-intercepts (include order):
v) y-intercept:
vi) the intervals where the function is positive / negative
Equations and Graphs of Poly. Functions
Ex. 2:
Sketch the function f(x) = 3(x - 1) (x + 3).
i) the degree:
ii) the sign of leading coefficient:
iii) end behaviour
x -, y _____
x , y _____
iv) x-intercepts (include order):
v) y-intercept:
vi) the intervals where the function is positive / negative
Equations and Graphs of Poly. Functions
Ex. 3:
Sketch the function f(x) = -(2x + 1) (x - 3).
the
the sign of leading coefficient:
end behaviour
,y
_____
,y
_____
y-intercept
vi) the intervals where the function is positive / negative
Equations and Graphs of Poly. Functions
Ex. 4:
Sketch the function f(x) = (x - 4) (x + 4).
the
the sign of leading coefficient:
end behaviour
,y
_____
,y
_____
y-intercept
vi) the intervals where the function is positive / negative
Equations and Graphs of Poly. Functions
Ex. 5:
Using the graph of the polynomial function, determine:
the least possible degree
leading coefficient
x-intercepts and factors of the
the intervals where the
function is positive and the
where the function
is negative
equation if the y-int is 6
Equations and Graphs of Poly. Functions
Equations and Graphs of Poly. Functions
EQUATIONS AND GRAPHS OF POLY FUNCTIONS
Sketching Polynomial Functions Summary
To sketch a polynomial function,
x Factor the polynomial fully, if it is not in factored form.
x Identify the degree. This will indicate the general shape of the curve.
x Look to see if the lead coefficient is positive or negative. This will help peg down the
shape and quadrants.
x Find the x-intercepts. Let y = 0 solve for x.
x Find the y-intercept. Let x = 0 solve for y.
x Plot the intercepts, note their ‘order’ and use the appropriate shapes to sketch the curve.
x Remember that if the variable or bracket has an even exponent, the curve “bounces” off
the intercept.
However, if the variable or bracket has an odd exponent, the curve passes through the
intercept in one of two ways:
x If the odd exponent is 1, then the curve passes straight through the axis.
x If the odd exponent is greater than 1, (i.e. 3, 5, 7. . . ) then, the curve bends
creating a slight shelf at the x-intercept.
ASSIGNMENT:
Figure out the degree, the sign of the lead coefficient, quadrants/shape, the roots/x-intercepts, and
the y-intercepts, Sketch each of the following:
a) y
x( x 2)( x 3)
b) y ( x 1)( x 3) 2
c) y x( x 3)( x 2)( x 4)
d)y
( x 2)3 ( x 3)
e) y
x( x 2) 2 ( x 2)
f )y
x 2 ( x 2)3
g) y
( x 1) 2 ( x 1)3 ( x 2)( x 2)
h) y
x3 x
i) y
x3 4 x 2 3x
j) y
( x 2 9)
Hint: h), i), and j) will need to be fully factored before you start.
DAY 4 - Transformations of Poly. Functions
f(x) = a[k(x - d)]
Transformations of Poly. Functions
f(x) = -2[-4(x + 6)]
f(x) = -2[-4(x + 6)]
f(x) = -2[-4(x + 6)]
Transformations of Poly. Functions
Transformations of Poly. Functions
(
,
)
Transformations of Poly. Functions
Ex.2:
Graph the following function,
(
,
)
DAY 5 - Slopes of Secants and Ave. Rate of Change
Slopes of Secants and Ave. Rate of Change
Slopes of Secants and Ave. Rate of Change
a) 1964 - 1974
1960
120 000
1964
110 000
1974
220 000
1985
190 000
1989
410 000
1996
260 000
2004
330 000
2007
390 000
b) 1989 - 1996
c) 1960 - 2007
Slopes of Secants and Ave. Rate of Change
Slopes of Secants and Ave. Rate of Change
DAY 6 - Slopes of Tangents and Instantenous Rate of Change
Slopes of Tangents and Instantenous Rate of Change
Slopes of Tangents and Instantenous Rate of Change
Interval
5.5
5.1
5.01
5.001
Rate of
Change

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