# What you should know by the end of the course

# What you should know by the end of the course

**1 Functions and Relations**

The definition of a function.

The definition of an inverse function. Detecting whether a function is
invertible.

Interpreting and evaluating statements like “ f (2)” or f ^{-1}(9) in practical
terms, with appropriate

units.

Domain and range of any function or transformation of a function of the
following types:

– linear

– exponential

– logarithmic

– sinusoidal

– power

– quadratic

– polynomial (possibly with the aid of a graphing calculator)

– rational.

Expressing intervals correctly, including whether the interval endpoints are
closed or open.

When given enough information, stating domains and ranges with exact values,
including whether

the interval ends are closed or open.

Identifying holes, breaks (function discontinuities), and asymptotes for the
functions (of their transformations)

in the above families.

Identifying contextually appropriate domain and range for a scenario.

Definition of “zeros” of a function.

Relationship between points on a graph of the function and the definition of a
function.

Finding an inverse function, given a graph, table, or formula.

Finding a formula for inverse function, using appropriate variables.

Graphs and formulas for combined functions .

Composing and combining functions as prescribed by a scenario .

Composing and combining functions symbolically.

Detecting whether a function is increasing, decreasing, concave up, or concave
down.

2 Linear Functions

Definition of linear function.

Find the formula and graph of line given any of the following pieces of
information:

– slope, vertical intercept

– table of values

– two points along the graph

– scenario that is modelled by a linear function.

Explain why assumption of constant rate of change means that a formula has the
form y = mx + b.

Recognize a linear function expressed in slope-intercept, point-slope, standard
forms. Convert between

any of those forms.

Find intersection between two linear functions.

Calculate perpendicular and parallel lines through a given point.

Identify when a scenario can be modelled by an linear function.

3 Exponential Functions

The definition of an exponential function as one of the form f (x) = a*b^{k} or f
(t) = a*e^{kt}, and being

able to convert between those two forms.

The definition of initial value, growth/decay factor, percentage growth/decay
rate, continuous growth/decay

rate.

Find the formula and graph of an exponential function given any of the following
pieces of information:

– initial value and growth /decay factor (or percentage growth/decay rate, or
continuous growth/decay

rate)

– table of values

– two points along the graph

– scenario that is modelled by an exponential function.

Identify when a scenario can be modelled by an exponential function.

Continuous growth/decay rate, e.

Compound interest .

Comparing the growth and decay rates of a collection of exponential functions.

4 Logarithmic Functions

Definition of log and ln.

Laws of logarithmic functions.

Double life, half life, and related concepts.

Logarithmic functions and exponential functions are inverses of each other.

5 Trigonometric Functions

Definition of a sinusoidal function

Find formula and graph of sinusoidal function given any of the following pieces
of information:

– amplitude, midline, period, phase shift, initial phase (angle)

– scenario roughly modelled by a sinusoidal function, such as a ferris wheel,
spring or coil, or the

weather

– table of values

Identify the amplitude, midline, period, phase shift, initial phase (angle) of a
sinusoidal function from

the formula or from a scenario.

Radians and degrees.

Special angles.

Algebraically solving equations such as “-5 sin(4x + 9) = 21” or “7 cos(2πx - 3)
- 2 = 3” for x in a

specified domain, in exact form.

Definition of sin ^{-1}and cos ^{-1} and the domains that belong to them.

6 Quadratic Functions

Definition of a quadratic function.

Vertex form (y = a(x - k) ^{2} + h), standard form (y = ax^{2} + bx + c), zero form (y
= a(x - r_{1})(x - r_{2})),

and how to convert between any pair of them.

Knowing the meaning of k and h in vertex form, and r_{1} and r_{2} in the zero form.

Knowing the relationship between a and whether the vertex is a maximum or
minimum.

Being able to explain when a vertex is a maximum or minimum. (Remember: if you
cite the graph,

then you need to draw the graph!)

Finding the formula and graph of a quadratic function, given any of the
following pieces of information:

– Three points along the graph

– Table of values

– Scenario modelled by a quadratic function

Solving a quadratic equation.

Completing the square .

The meaning of the coordinates of the vertex, when given a scenario that can be
modelled by a

quadratic function.

7 Power Functions

Definition of power function.

Definition of power, constant of proportionality

Find the formula and graph of power function given any of the following pieces
of information:

– two points along the graph

– scenario modelled by power function

– table of values

– power, constant of proportionality

Understanding the difference between power functions whose powers are negative,
less than 1, equal

to 1, positive and greater than 1.

8 Polynomial Functions

Definition of a polynomial function.

Definition of “factored form” (e.g., y = 5(x -1)^{2}(x -5)^{4}(x +3)) and standard
form for a polynomial.

Definition of degree, term, coefficients , constant term, leading term.

Finding the formula for a polynomial given a graph, and understanding the
relationship between the

formula and:

– zeros that locally look like “chairs”

– zeros that locally look like “bumps”

– zeros that locally look like a line

– any additionally specified coordinate points on the graph

Understanding relationship between long-run behaviour of polynomial and the
leading term.

Understanding the difference between the short-run behaviour of a polynomial and
the long-run

behaviour of a polynomial.

9 Rational Functions

Definition of rational function.

Understanding relationship between long-run behaviour of rational functions and
leading terms in

polynomials in numerator and denominator .

Finding the formula for a rational function give a graph, and understanding the
relationship between

the formula and:

– vertical asymptotes

– horizontal asymptotes

– vertical intercepts

– horizontal intercepts

– holes

– any additionally specified coordinate points on the graph

Transformations of functions of the form f (x) = 1/x and f (x) = 1/x^{2} , and writing them as rational

functions. If you are told that a function g(x) is a transformation of f (x) = 1/x or f (x) = 1/x^{2} , you can

identify which transformations you need, and express g(x) in the form g(x) = k f
(a(x + b/a)) +e.

(See Note in ”Top Ways NOT to Lose Points” on transformations.)

10 Long-Run Behaviour

Understanding the relative long-run behaviours of power, exponential, and
logarithmic functions,

and how it relates to the long-run behaviours of functions such as f (x) = g(x)/h(x) , when g and h are

power, exponential, or logarithmic functions or combinations thereof.

Understanding the long-run behaviour of power functions and how it relates to
the power.

Understanding the long-run behaviour of exponential functions and how it relates
to the growth

factor.

11 Direct and Inverse Proportionality

Definition of direct and inverse proportionality

Understanding the meaning of phrases such as “A function is directly
proportional to ... ” and “A

function is inversely proportional to ...”.

Definition of constant of proportionality.

12 Transformations of Functions and Their Graphs

Definition of horizontal and vertical stretches/compressions, reflections, and
shifts.

Expressing the transformations in function notation. (For example: “the function
f (x - a) is a horizontal

shift of +a from f (x).”)

The correct order for successive transformations.

Correctly using the terms stretch vs. compressions (mimic the usage on p. 212
and p. 221 in your

textbook).

Being able to interpret transformations from a scenario.

Relationship between transformations and placement of specified points on
graphs, asymptotes, and

long-run behaviour.

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