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MCR3U Grade 11 Mathematics Full Course Notes. 4.0 GPA
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MCR3U Grade 11 Mathematics Full Course Notes. 4.0 GPA
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Key Notes (Retrieved from Math Journal):-
Unit 1: Functions and Transformations
otes:
N
A relation is a rule for pairing 2 numbers to form a set of ordered pairs.
There are 6 ways to describe relationships:
- Using ordered pairs
- Using words
- Using an equation
- Table of Values
- Using a graph
- Using mapping diagrams
functionis a relation where each input/independantx value is paired
A
with exactly 1 out/dependent y value. → One x value cannot have
multiple y-values.
To confirm if a relation is a function we can use the vertical line test.
Domain:a set of all possible values of the independentvariable (x) for a
relation. {X ϵ R | X __ __) AKA. X belongs / is an element of all real
numbers such that Y obeys __ __ Restriction(s)
Range:a set of all possible values of the dependentvariable (y) for a
relation. {Y ϵ R | Y __ __) AKA. Y belongs / is an element of all real
numbers such that X obeys __ __ Restriction(s)
Domain and Range are represented using Set Notation
,Common Parent Functions:
Linear Function:
(𝑥) = 𝑥
𝑓
Domain: {X ϵ R}
Range: {Y ϵ R}
Quadratic Function:
2
(𝑥) = 𝑥
𝑓
Domain: {X ϵ R}
Range: {Y ϵ R |Y ≥ 0}
Square Root Function:
(𝑥) = 𝑥
𝑓
Domain: {X ϵ R |X ≥ 0}
Range: {Y ϵ R |Y ≥ 0}
Absolute Value Function:
(𝑥) = |𝑥|
𝑓
Domain: {X ϵ R}
Range: {Y ϵ R |Y ≥ 0}
Rational Function:
1
𝑓(𝑥) = 𝑥
omain: {X ϵ R | X ≠ 0}
D
Range: {Y ϵ R |Y ≠ 0}
,Special Functions:
Ceiling Function:
(𝑥) = 𝐼𝑛𝑡𝑒𝑔𝑒𝑟𝑅𝑜𝑢𝑛𝑑𝑈
𝑓 𝑝:
|𝑥|
Domain: {X ϵ R}
Range: {Y ϵ R}
Floor Function:
(𝑥) = 𝐼𝑛𝑡𝑒𝑔𝑒𝑟𝑅𝑜𝑢𝑛𝑑𝐷
𝑓 𝑜𝑤𝑛:
|𝑥|
Domain: {X ϵ R}
Range: {Y ϵ R}
Signum Function:
(𝑥) = 𝑠𝑔𝑛(𝑥)
𝑓
Positive Integer returns +1, Negative integer returns -1, and a Zero
returns 0
Domain: {X ϵ R}
Range: {Y ϵ R | -1 ≥ Y ≤ 1}
irichlet Function:
D
The Dirichlet Function is an indicator function that takes all x values that
belong to a set and returns 1 as the y-value. All other x values return 0.
.g Prime Number Dirichlet Function:
E
D(3) = 1 (True)
D(4) = 0 (False)
D(11) = 1 (True)
, ase functions are the simplest form of its type of function. Consequently,
B
transformations can be performed on them.
Types of Transformations:
Order = RST: Reflection, Stretch, Translation
1 . Identify the Base Function
2. Identify the transformations applied
𝑦 = − 𝑎𝑓[− 𝑘( 𝑥 − 𝑑)] + 𝑐
− reflection on the x axis
a vertical stretch (Stretched by a factor of a. Pulled/Compressed From
Above)
− reflection on the y axis
k horizontal stretch (Stretched by a factor of 1/k. Pulled/Compressed From
the Side)
d left or right (+d moves the points left, -d moves the points right)
c up or down (+c moves the points up, -c moves the points down)
ffects Y
A
Affects X
ased on the transformations enacted by a function on its X and Y
B
variables, we can create equations to solve for its transformed X and Y
variables individually.
Graph & Highlight any Asymptotes!
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