K. Stewart
MHF4U Unit Outline
Chapter 1: Polynomial Functions
Throughout this chapter, you will explore how the curves represented by polynomial functions are applied in various
design-related fields such as civil engineering, architectural design, computer graphics design, interior design, and
landscape architecture.
By the end of this chapter, you will be assessed on your ability to:
C1.0
D1.0
identify and describe some key features of polynomial functions, and make connections between the numeric,
graphical, and algebraic representations of polynomial functions.
demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and
graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate
of change of a function at a given point.
Section
1.1 Power
Functions
1.2
Characteristics
of Polynomial
Functions
Learning goals
What are the key features of the graphs of
power function?
How can I recognize polynomial functions?
What is the connection between the graphs
and equations of power functions?
Vocab: power, exponent, power function,
polynomial function, trigonometric function,
rational function, exponential function,
square root function, coefficient, degree,
leading coefficient, constant term, linear,
quadratic, cubic, quartic, quintic,
polynomial, key features, domain, range,
end behaviour, symmetry (line symmetry,
point symmetry), axis of symmetry,
intercepts
Notation: x  , x  – 
interval notation –  < x  0
set notation (– , 0]
graphical notation (on a number line)
domain notation e.g. {x| x  }
range notation e.g. {y| y > 0, y  }
Key equation: Power function is of the form
f(x) = axn where n is a whole number
Polynomial function is of the form
f(x)= an xn +an-1 xn-1 +an-2 xn-2 +…+a1 x1 +a0
where coefficients are real numbers and n is
a whole number
What are the key features of the graphs of
polynomial functions?
What is the relationship between finite
differences and the equation of a
polynomial function?
What is the connection between the graphs,
equations and tables of values of
polynomial functions?
Vocab: local minimum (or maximum), absolute
(or global) minimum (or maximum),
optimal value, finite difference, finite
difference table, factorial
Notation: n! is read n factorial and is the
product (n)(n – 1) (n – 2)…(2)(1)
Homework
Pg 11 #1 – 4 optional
Pg 12 #6, 7, 8, 13 mandatory
Skills: match graph with corresponding power function,
name function by degree, identify power functions from their
equations, use interval notation, set notation and graphical
notation interchangeably, identify key features of power
functions (domain and range, end behaviour, symmetry,
intercepts, degree, positive or negative leading coefficient),
solve problems involving power functions
Pg 26 #1 – 4, 12 optional
Pg 27 #5 – 8, 11 mandatory
Skills: match graph with corresponding polynomial function,
name function by degree, identify polynomial functions from
their equations, use interval notation, set notation and
graphical notation interchangeably, identify key features of
polynomial functions (domain and range, end behaviour,
symmetry, intercepts, degree, positive or negative leading
coefficient), solve problems involving polynomial functions,
determine value of finite difference using a difference table
and algebraically, use a regression analysis on a graphing
calculator to determine the equation of best fit.
(Continued next page)
K. Stewart
1.3 Equations
and Graphs of
Polynomial
Functions
1.4
Transformations
1.5 Slopes of
Secants and
Average Rate of
Change
1.6 Slopes of
Tangents and
Instantaneous
Rate of Change
What is the connection between the factored
form of a polynomial function and its
graph?
How can I sketch graphs of polynomial
functions from the equations?
How is symmetry represented in the
equation of a polynomial function?
Vocab: order of an intercept, even function,
odd function, constants are coefficients of
x0,
Key equations: even function f(x) = f(–x)
odd function –f(x) = f(–x)
(day 1)
Pg 39 #1, 2 optional
Pg 40 # 9, 11, 12 mandatory
What are the roles of a, k, d, and c in
polynomial functions of the form
y = a[k(x-d)]n+c where n  N?
How can I describe transformations from an
equation and use them to sketch a graph?
How can I determine an equation give the
graph of a transformed function?
Vocab: transformation, reflection, translation,
stretch, compression, opening
Notation: y = a[k(x-d)] n+c
How can I connect average rate of change
and slope?
How can I calculate and interpret average
rates of change from any given
representation?
Vocab: average rate of change, slope, secant,
radius
What is the connection between the slopes of
secants, the slope of a tangent and the
instantaneous rate of change?
How can I estimate an instantaneous rate of
change from any representation?
Vocab: instantaneous rate of change, tangent,
velocity
Pg 49 #1 – 7 optional
Pg 50 #8, 10, 11, 12 mandatory
Skills: write an equation of a polynomial function from a
description of its key features, identify zeros of an equation,
sketch a graph from an equation
(day 2)
Pg 39 #3 – 5 optional
Pg 40 #6, 7, 8, 10, 15 mandatory
Skills: write an equation from a graph of a polynomial
function, identify symmetry of graph from equation,
determine whether polynomial function is an even function,
odd function or neither.
Skills: applying transformations to sketch a graph, identify
and describe transformations from a graph or an equation,
determine equation from graph using transformations,
solving for parameters given graph or value of function at a
point.
Pg 62 #1 – 5, 9, 12 optional
Pg 62 #6, 7, 8, 10 mandatory
Skills: connect rate of change to slope, calculate and
interpret average rates of change from a graph, table of
values or an equation.
Pg 71 # 1, 2, 7, 8 optional
Pg 71 #3, 4, 5, 10, 11 mandatory
Skills: describe relationship between slope of secants and the
slope of a tangent, estimate instantaneous rate of change of a
point from a graph, table of values or an equation,
differentiate between the rate of change at a point and the
value of the function at that point.
The Chapter will wrap up with a review, the Chapter 1 Task and a written test. Class discussions, class work, homework
and quizzes will help you to determine how well you understand the course material in preparation for the summative
assessments.
Extra help is available with me upon request.
There is free tutoring and instructional videos available on the class moodle.