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*bk or f
(t) = a*ekt, 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 -1and 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 = ax2 + bx + c), zero form (y
= a(x - r1)(x - r2)),
and how to convert between any pair of them.
Knowing the meaning of k and h in vertex form, and r1 and r2 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/x2 , 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/x2 , 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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