Course Description
This course introduces the mathematical concept of the function by extending students' experiences with linear and quadratic relations. Students will investigate properties of discrete and continuous functions, including trigonometric and exponential functions; represent functions numerically, algebraically, and graphically; solve problems involving applications of functions; investigate inverse functions; and develop facility in determining equivalent algebraic expressions. Students will reason mathematically and communicate their thinking as they solve multi-step problems.
Overall Provincial Curriculum Expectations
See Ontario math curriculum for grades 11 and 12 for
details.
A: Characteristics of Functions
- demonstrate an understanding of functions, their representations, and their inverses, and make
connections between the algebraic and graphical representations of functions using transformations;
- determine the zeros and the maximum or minimum of a quadratic function, and solve problems
involving quadratic functions, including problems arising from real-world applications;
- demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and
rational expressions.
B: Exponential Functions
- evaluate powers with rational exponents, simplify expressions containing exponents, and describe
properties of exponential functions represented in a variety of ways;
- make connections between the numeric, graphical, and algebraic representations of exponential
functions;
- identify and represent exponential functions, and solve problems involving exponential functions,
including problems arising from real-world applications.
C: Discrete Functions
- demonstrate an understanding of recursive sequences, represent recursive sequences in a variety of
ways, and make connections to Pascal’s triangle;
- demonstrate an understanding of the relationships involved in arithmetic and geometric sequences
and series, and solve related problems;
- make connections between sequences, series, and financial applications, and solve problems involving
compound interest and ordinary annuities.
D: Trigonometric Functions
- determine the values of the trigonometric ratios for angles less than 360º; prove simple trigonometric
identities; and solve problems using the primary trigonometric ratios, the sine law, and the cosine law;
- demonstrate an understanding of periodic relationships and sinusoidal functions, and make
connections between the numeric, graphical, and algebraic representations of sinusoidal functions;
- identify and represent sinusoidal functions, and solve problems involving sinusoidal functions,
including problems arising from real-world applications.
Specific Provincial Curriculum Expectations
A1: Representing Functions
- 1.1 explain the meaning of the term function, and
distinguish a function from a relation that is
not a function, through investigation of linear
and quadratic relations using a variety of representations
(i.e., tables of values, mapping diagrams,
graphs, function machines, equations)
and strategies (e.g., identifying a one-to-one
or many-to-one mapping; using the verticalline
test)
- 1.2 represent linear and quadratic functions using
function notation, given their equations, tables
of values, or graphs, and substitute into and
evaluate functions [e.g., evaluate $f(\frac{1}{2})$, given
$f(x) = 2x^2 + 3x – 1$]
- 1.3 explain the meanings of the terms domain
and range, through investigation using numeric,
graphical, and algebraic representations of
the functions $f(x)=x,~x^2,~\sqrt{x},~ \frac{1}{x}$;
describe the domain and range of a function appropriately (e.g., for $y=x^2+1$, the domain is the set
of all real numbers and the range is the set $y\geq 1$); and explain any restrictions on the
domain and range in contexts arising from real-world applications
- 1.4 relate the process of determining the inverse
of a function to their understanding of
reverse processes (e.g., applying inverse
operations)
- 1.5 determine the numeric or graphical representation
of the inverse of a linear or quadratic
function, given the numeric, graphical, or
algebraic representation of the function, and
make connections, through investigation
using a variety of tools (e.g., graphing technology,
Mira, tracing paper), between the
graph of a function and the graph of its
inverse (e.g., the graph of the inverse is the
reflection of the graph of the function in the
line $y = x$)
- 1.6 determine, through investigation, the relationship
between the domain and range of a function
and the domain and range of the inverse
relation, and determine whether or not the
inverse relation is a function
- 1.7 determine, using function notation when
appropriate, the algebraic representation of
the inverse of a linear or quadratic function,
given the algebraic representation of the
function and make
connections, through investigation using a
variety of tools (e.g., graphing technology,
Mira, tracing paper), between the algebraic
representations of a function and its inverse
(e.g., the inverse of a linear function involves
applying the inverse operations in the reverse
order)
- 1.8 determine, through investigation using
technology, the roles of the parameters
$a, k, d, c$ in functions of the form
$y = af (k(x – d)) + c$, and describe these roles
in terms of transformations on the graphs
of $f(x)=x,~x^2,~\sqrt{x},~ \frac{1}{x}$ (i.e., translations; reflections in the
axes; vertical and horizontal stretches and
compressions to and from the x- and y-axes)
- 1.9 sketch graphs of $y = af (k(x – d)) + c$
by applying one or more transformations
to the graphs of $f(x)=x,~x^2,~\sqrt{x},~ \frac{1}{x}$, and state the domain and range of the transformed functions
A2: Solving Problems Involving Quadratic Functions
- 2.1 determine the number of zeros (i.e.,
x-intercepts) of a quadratic function, using
a variety of strategies (e.g., inspecting graphs;
factoring; calculating the discriminant)
- 2.2 determine the maximum or minimum value
of a quadratic function whose equation is
given in the form $f(x) = ax^2 + bx + c$, using
an algebraic method (e.g., completing the
square; factoring to determine the zeros and
averaging the zeros)
- 2.3 solve problems involving quadratic functions
arising from real-world applications and
represented using function notation
- 2.4 determine, through investigation, the transformational
relationship among the family of
quadratic functions that have the same zeros,
and determine the algebraic representation of
a quadratic function, given the real roots of
the corresponding quadratic equation and a
point on the function
- 2.5 solve problems involving the intersection of
a linear function and a quadratic function
graphically and algebraically (e.g., determine
the time when two identical cylindrical water
tanks contain equal volumes of water, if one
tank is being filled at a constant rate and the
other is being emptied through a hole in the
bottom)
A3: Determining Equivalent Algebraic Expressions
- 3.1 simplify polynomial expressions by adding,
subtracting, and multiplying
- 3.2 verify, through investigation with and
without technology, that $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$, $a,b\geq 0$, and use this relationship to simplify radicals
and radical expressions obtained by adding, subtracting,
and multiplying
- 3.3 simplify rational expressions by adding,
subtracting, multiplying, and dividing, and
state the restrictions on the variable values
- 3.4 determine if two given algebraic expressions
are equivalent (i.e., by simplifying; by
substituting values).
B1: Representing Exponential Functions
- 1.1 graph, with and without technology, an exponential
relation, given its equation in the form
$y = a^x$ ($a > 0, ~a\neq 1$), define this relation as the
function $f(x) = a^x$, and explain why it is a
function
- 1.2 determine, through investigation using a
variety of tools (e.g., calculator, paper and
pencil, graphing technology) and strategies
(e.g., patterning; finding values from a graph;
interpreting the exponent laws), the value of
a power with a rational exponent (i.e. $x^{m/n}$ where $x>0$ and $m$ and $n$ are integers)
- 1.3 simplify algebraic expressions containing
integer and rational exponents, and evaluate numeric
expressions containing integer and rational
exponents and rational bases
- 1.4 determine, through investigation, and
describe key properties relating to domain
and range, intercepts, increasing/decreasing
intervals, and asymptotes (e.g., the domain
is the set of real numbers; the range is the
set of positive real numbers; the function
either increases or decreases throughout its
domain) for exponential functions represented
in a variety of ways [e.g., tables of values,
mapping diagrams, graphs, equations of the
form $f(x) = a^x$, function
machines]
B2: Connecting Graphs and Equations of Exponential Functions
- 2.1 distinguish exponential functions from linear
and quadratic functions by making comparisons
in a variety of ways (e.g., comparing
rates of change using finite differences in
tables of values; identifying a constant ratio in
a table of values; inspecting graphs; comparing
equations)
- 2.2 determine, through investigation using technology,
the roles of the parameters $a, k, d,
c$ in functions of the form $y = af (k(x – d)) + c$,
and describe these roles in terms of transformations
on the graph of $f(x) = a^x$
(i.e., translations; reflections in the axes; vertical
and horizontal stretches and compressions
to and from the x- and y-axes)
- 2.3 sketch graphs of $y = af (k(x – d)) + c$ by
applying one or more transformations
to the graph of $f(x) = a^x$,
and state the domain and range of the
transformed functions
- 2.4 determine, through investigation using technology,
that the equation of a given exponential
function can be expressed using different bases
[e.g., $f(x) = 9^x$ can be expressed as $f(x) =3^{2x}$ ],
and explain the connections between the
equivalent forms in a variety of ways (e.g.,
comparing graphs; using transformations;
using the exponent laws)
- 2.5 represent an exponential function with an
equation, given its graph or its properties
B3: Solving Problems Involving Exponential Functions
- 3.1 collect data that can be modelled as an exponential
function, through investigation with
and without technology, from primary sources,
using a variety of tools (e.g., concrete materials
such as number cubes, coins; measurement
tools such as electronic probes), or from
secondary sources (e.g., websites such as
Statistics Canada, E-STAT), and graph
the data
- 3.2 identify exponential functions, including
those that arise from real-world applications
involving growth and decay (e.g., radioactive
decay, population growth, cooling rates,
pressure in a leaking tire), given various
representations (i.e., tables of values, graphs,
equations), and explain any restrictions that
the context places on the domain and range
(e.g., ambient temperature limits the range
for a cooling curve)
- 3.3 solve problems using given graphs or
equations of exponential functions arising
from a variety of real-world applications
(e.g., radioactive decay, population growth,
height of a bouncing ball, compound interest)
by interpreting the graphs or by substituting
values for the exponent into the equations
C1: Representing Sequences
- 1.1 make connections between sequences and
discrete functions, represent sequences using
function notation, and distinguish between a
discrete function and a continuous function
[e.g., $f(x) = 2x$, where the domain is the set of
natural numbers, is a discrete linear function
and its graph is a set of equally spaced points;
$f(x) = 2x$, where the domain is the set of real
numbers, is a continuous linear function and
its graph is a straight line]
- 1.2 determine and describe (e.g., in words; using
flow charts) a recursive procedure for generating
a sequence, given the initial terms
(e.g., 1, 3, 6, 10, 15, 21, $\cdots$), and represent
sequences as discrete functions in a variety
of ways (e.g., tables of values, graphs)
- 1.3 connect the formula for the nth term of a
sequence to the representation in function
notation, and write terms of a sequence given
one of these representations or a recursion
formula
- 1.4 represent a sequence algebraically using a
recursion formula, function notation, or the
formula for the nth term [e.g., represent 2, 4,
8, 16, 32, 64,$\cdots$ as $t_1 = 2; t_n = 2t_{n – 1}$, as $f(n)=2^n$ or as $t_n=2^n$ where $n$ is a natural number], and describe the information
that can be obtained by inspecting
each representation (e.g., function notation
or the formula for the nth term may show
the type of function; a recursion formula
shows the relationship between terms)
- 1.5 determine, through investigation, recursive
patterns in the Fibonacci sequence, in related
sequences, and in Pascal’s triangle, and
represent the patterns in a variety of ways
(e.g., tables of values, algebraic notation)
- 1.6 determine, through investigation, and
describe the relationship between Pascal’s
triangle and the expansion of binomials,
and apply the relationship to expand binomials
raised to whole-number exponents
C2: Investigating Arithmetic and Geometric Sequences and Series
- 2.1 identify sequences as arithmetic, geometric,
or neither, given a numeric or algebraic
representation
- 2.2 determine the formula for the general
term of an arithmetic sequence [i.e.,
$t_n = a + (n –1)d $] or geometric sequence
(i.e., $t_n = ar^{n – 1}$), through investigation
using a variety of tools (e.g., linking cubes,
algebra tiles, diagrams, calculators) and
strategies (e.g., patterning; connecting the
steps in a numerical example to the steps in
the algebraic development), and apply the
formula to calculate any term in a sequence
- 2.3 determine the formula for the sum of an
arithmetic or geometric series, through investigation
using a variety of tools (e.g., linking
cubes, algebra tiles, diagrams, calculators)
and strategies (e.g., patterning; connecting
the steps in a numerical example to the steps
in the algebraic development), and apply
the formula to calculate the sum of a given
number of consecutive terms
- 2.4 solve problems involving arithmetic and geometric
sequences and series, including those
arising from real-world applications
C3: Solving Problems Involving Financial Applications
- 3.1 make and describe connections between
simple interest, arithmetic sequences, and
linear growth, through investigation with
technology (e.g., use a spreadsheet or
graphing calculator to make simple interest
calculations, determine first differences in
the amounts over time, and graph amount
versus time)
- 3.2 make and describe connections between
compound interest, geometric sequences,
and exponential growth, through investigation
with technology (e.g., use a spreadsheet
to make compound interest calculations,
determine finite differences in the amounts
over time, and graph amount versus time)
- 3.3 solve problems, using a scientific calculator,
that involve the calculation of the amount,
$A$ (also referred to as future value, $FV$),
the principal, $P$ (also referred to as
present value, $PV$), or the interest rate
per compounding period, $i$, using the
compound interest formula in the form
$A = P(1 + i)^n$ [or $FV = PV(1 + i)^n$ ]
- 3.4 determine, through investigation using
technology (e.g., scientific calculator, the
TVM Solver on a graphing calculator, online
tools), the number of compounding periods, n,
using the compound interest formula in the
form $A = P(1 + i)^n$ [or $FV = PV(1 + i)^n$ ];
describe strategies (e.g., guessing and checking;
using the power of a power rule for
exponents; using graphs) for calculating this
number; and solve related problems
- 3.5 explain the meaning of the term annuity, and
determine the relationships between ordinary
simple annuities (i.e., annuities in which payments
are made at the end of each period, and
compounding and payment periods are the
same), geometric series, and exponential
growth, through investigation with technology
(e.g., use a spreadsheet to determine and
graph the future value of an ordinary simple
annuity for varying numbers of compounding
periods; investigate how the contributions of
each payment to the future value of an ordinary
simple annuity are related to the terms
of a geometric series)
- 3.6 determine, through investigation using
technology (e.g., the TVM Solver on a graphing
calculator, online tools), the effects of
changing the conditions (i.e., the payments,
the frequency of the payments, the interest
rate, the compounding period) of ordinary
simple annuities (e.g., long-term savings
plans, loans)
- 3.7 solve problems, using technology (e.g., scientific
calculator, spreadsheet, graphing calculator),
that involve the amount, the present
value, and the regular payment of an ordinary
simple annuity (e.g., calculate the total
interest paid over the life of a loan, using a
spreadsheet, and compare the total interest
with the original principal of the loan)
D1: Determining and Applying Trigonometric Ratios
- 1.1 determine the exact values of the sine, cosine,
and tangent of the special angles: 0º, 30º, 45º,
60º, and 90º
- 1.2 determine the values of the sine, cosine, and
tangent of angles from 0º to 360º, through
investigation using a variety of tools (e.g.,
dynamic geometry software, graphing tools)
and strategies (e.g., applying the unit circle;
examining angles related to special angles)
- 1.3 determine the measures of two angles from
0º to 360º for which the value of a given
trigonometric ratio is the same
- 1.4 define the secant, cosecant, and cotangent
ratios for angles in a right triangle in
terms of the sides of the triangle (e.g.,
$\sec A =\frac{hypotenuse}{adjacent} $), and relate these ratios
to the cosine, sine, and tangent ratios (e.g.,
$\sec A =\frac{1}{\cos A} $)
- 1.5 prove simple trigonometric identities, using
the Pythagorean identity $\sin^2 x+\cos^2 x=1$; the quotient identity $\tan x=\frac{\sin x}{\cos x}$; and the reciprocal identities
- 1.6 pose problems involving right
triangles and oblique triangles in twodimensional
settings, and solve these and
other such problems using the primary
trigonometric ratios, the cosine law, and
the sine law (including the ambiguous case)
- 1.7 pose problems involving right triangles and
oblique triangles in three-dimensional settings,
and solve these and other such problems
using the primary trigonometric ratios,
the cosine law, and the sine law
D2: Connecting Graphs of Equations and Sinusoidal Functions
- 2.1 describe key properties (e.g., cycle, amplitude,
period) of periodic functions arising from
real-world applications (e.g., natural gas
consumption in Ontario, tides in the Bay
of Fundy), given a numeric or graphical
representation
- 2.2 predict, by extrapolating, the future behaviour
of a relationship modelled using a numeric or
graphical representation of a periodic function
(e.g., predicting hours of daylight on a particular
date from previous measurements; predicting
natural gas consumption in Ontario from
previous consumption)
- 2.3 make connections between the sine ratio and
the sine function and between the cosine ratio
and the cosine function by graphing the
relationship between angles from 0º to 360º
and the corresponding sine ratios or cosine
ratios, with or without technology (e.g., by
generating a table of values using a calculator;
by unwrapping the unit circle), defining this
relationship as the function $f(x) =\sin x$ or
$f(x) =\cos x$, and explaining why the relationship
is a function
- 2.4 sketch the graphs of $f(x) =\sin x$ and
$f(x) =\cos x$ for angle measures expressed
in degrees, and determine and describe
their key properties (i.e., cycle, domain, range,
intercepts, amplitude, period, maximum
and minimum values, increasing/decreasing
intervals)
- 2.5 determine, through investigation using technology,
the roles of the parameters $a, k, d,
c$ in functions of the form $y =af (k(x – d)) + c$,
where $f(x) =\sin x$ or $f(x) =\cos x$ with angles
expressed in degrees, and describe these roles
in terms of transformations on the graphs of
$f(x) =\sin x$ and $f(x) =\cos x$ (i.e., translations;
reflections in the axes; vertical and horizontal
stretches and compressions to and from the
x- and y-axes)
- 2.6 determine the amplitude, period, phase
shift, domain, and range of sinusoidal
functions whose equations are given in
the form $y = a\sin(k(x – d)) + c$ or
$y = a\cos(k(x – d)) + c$
- 2.7 sketch graphs of $y = af (k(x – d)) + c$ by
applying one or more transformations to the
graphs of $f(x) =\sin x$ and $f(x) =\cos x$, and state
the domain and range of the transformed
functions
- 2.8 represent a sinusoidal function with an
equation, given its graph or its properties
D3: Solving Problems Involving Trigonometric Functions
- 3.1 collect data that can be modelled as a sinusoidal
function (e.g., voltage in an AC circuit,
sound waves), through investigation with
and without technology, from primary
sources, using a variety of tools (e.g., concrete
materials, measurement tools such as motion
sensors), or from secondary sources (e.g.,
websites such as Statistics Canada, E-STAT),
and graph the data
- 3.2 identify periodic and sinusoidal functions,
including those that arise from real-world
applications involving periodic phenomena,
given various representations (i.e., tables of
values, graphs, equations), and explain any
restrictions that the context places on the
domain and range
- 3.3 determine, through investigation, how sinusoidal
functions can be used to model periodic
phenomena that do not involve angles
- 3.4 predict the effects on a mathematical model
(i.e., graph, equation) of an application
involving periodic phenomena when the
conditions in the application are varied
(e.g., varying the conditions, such as speed
and direction, when walking in a circle in
front of a motion sensor)
- 3.5 pose problems based on applications involving
a sinusoidal function, and solve these and
other such problems by using a given graph
or a graph generated with technology from
a table of values or from its equation