mcr3u 1.1 relations and functions date:
TRANSCRIPT
1
MCR3U 1.1 Relations and Functions Date: ______________
Relation: a relationship between sets of information. ie: height and time of a ball in the air.
Relations can be represented in terms of:
Equations:
y = 2x - 3
Words:
y is equal to two
times x minus 3
Graph:
Table of Values: Mapping Diagram:
Function: A function is a relation in which each value of the independent variable (x-value) corresponds to
exactly one value of the dependent variable (y- value). *Each x-value must produce only one y-value.*
Domain: the set of all values of the independent variable of a relation (x-values)
Range: the set of all values of the dependent variable of a relation (y-values)
State the domain and range for these relations.
a) b)
Determine which the following graphs represent a function.
2
Determine whether the following sets are functions
a) {(1, -2), (2, -4), (3, -6), (4, -8)} b) {(2, -2), (3, -4), (2, -5), (-1, -7)} c) {(-2, -3), (2, -3), (4, -1), (-4, -1)}
Determine if the following equations represent a function
a) 2π₯ + 3π¦ = 7 b) π¦ = 2π₯2 + π₯ β 5 c) π₯2 + π¦2 = 25 d) π₯ = π¦2 β 4
1.1 Assignment: p.10 #1, 2, 4, 6, 7, 11-13 p.2 #4, 7
-----------------------------------------------------------------------------------------------------------------------------
MCR3U 1.2 Function Notation Date: __________
To represent functions using function notation, we use notations such as f(x) and g(x).
The notation f(x) is read βf" of "xβ or βf" at "xβ
The symbol f(x) represents the dependent variable (y-value).
π(3) means the y-value, when x = 3
β―
Old notation: y = 25x + 200 Function notation: f(x) = 25x + 200 or C(n)= 25n + 200
Ex.1: Given the function (π₯) = π₯2 β 4 , determine:
a) π(1) b) π(β2) c) π₯ ππ π(π₯) = 5
3
Ex.2: Consider the functions: π(π₯) = π₯2 β 2π₯ π(π₯) = β3π₯ β 1
a) Show that π(1) < π(β2)
b) Determine each of the following:
π(1) + π(3) 2π(5)
π(β2π) π(π + 1) β π(π)
1.2 Assignment: P.22 #2 4a 5cd 6 7 8b(ii,iii,v,vi) 9b(iv,vi) 10 11bc 12 15 16c 17
4
MCR3U 1.3 Exploring Properties of Parent Functions Date: _____________
Three new functions:
1. Root Function π¦ = βπ₯
2. Absolute Value Function π¦ = |π₯|
3. Rational Function π¦ =1
π₯
1) ROOT Function: π¦ = βπ₯
Table of Values: Graph:
x y
Domain:
Range:
2) ABSOLUTE VALUE Function: π¦ = |π₯|
Table of Values: Graph:
Domain:
Range:
x y
5
3) RATIONAL Function: π¦ =1
π₯ , π₯ β 0
Table of Values: Graph:
x y
Domain: Equation of Asymptotes:
Range: Horizontal: Vertical:
1.3 Assignment: p.28 #1-3 & p. 558 #1d, 2be, 3d & p.556 #5bc
-----------------------------------------------------------------------------------------------------------------------------
MCR3U 1.4 Determining the Domain and Range of a Function Date: ____________
Recall: The domain is the set of all first coordinates (x-values) of the relation.
The range is the set of all second coordinates (y-values) of the relation.
Ex.1: Determine the domain and range
a) π¦ = βπ₯ + 4 b) {(-2, 2), (0, -1), (1, 0), (3, -1)}
6
c) d)
Ex2. State the domain and range for the following functions. Include a sketch.
a) π(π₯) = β3π₯ + 1 b) π(π₯) = 4(π₯ + 1)2 β 3
c) β(π₯) = βπ₯ β 2 d) π(π₯) =1
π₯+5
1.4 Assignment: p.35 #1-4, 7, 11, 14a
7
MCR3U 1.6 Transformations (shifts & reflections) Date: ____________
Transformations: A change made to a figure or a relation such that it is shifted or changed in shape.
- Translations, reflections and stretches/compressions are types of transformations.
Translations: A transformation that results in a shift (up, down, left, right) of the original figure without
changing its shape.
HORIZONTAL AND VERTICAL TRANSLATIONS
1) Vertical Translation of c units :
* The graph of the function g(x) = f(x) + c.
β―β when c is positive, the translation is UP
by c units.
β―β when c is negative, the translation is DOWNβ―
by c units.
2) Horizontal Translation of d units :
* The graph of the function g(x) = f(x β d).
β―β when d > 0, the translation is to the RIGHTβ―
by d units.
β―β when d < 0, the translation is to the LEFTβ―
by d units.
Example 1: Given the functions f(x) graphed below, sketch the graph of g(x) on the same grid.
a) g(x) = f(x) - 5 β― b) g(x) = f(x + 3)
8
Example 2: Graph the following parent functions and their transformed functions. Describe the
transformations that occurred. State the domain and range of the transformed functions.
a) π(π₯) = βπ₯ and π(π₯) = βπ₯ β 2 + 4
Transformations:
Domain:
Range:
b) π(π₯) =1
π₯ and π(π₯) =
1
π₯+3β 2
Transformations:
Domain:
Range:
Asymptotes:
9
1 2 3 4 5 6 7 8 9 10β1β2β3β4β5β6β7β8β9β10 x
1
2
3
4
5
6
7
8
9
10
β1
β2
β3
β4
β5
β6
β7
β8
β9
β10
y
REFLECTIONS OF FUNCTIONS
Reflection : A transformation in which a figure is reflected over a reflection line.
1) Reflection in the xβaxis or Vertical Reflection : The graph of g(x) = -f(x)
Graph the following functions along with their base functions.
a) π(π₯) = |π₯| and π(π₯) = β|π₯| b) π(π₯) = βπ₯ and π(π₯) = ββπ₯
2) Reflection in the yβaxis or Horizontal Reflection : The graph of g(x) = f(-x)
Graph the following functions along with their base functions.
a) π(π₯) = βπ₯ and π(π₯) = ββπ₯ b) π(π₯) =1
π₯ and π(π₯) =
1
βπ₯
1 2 3 4 5 6β1β2β3β4β5β6 x
1
2
3
4
5
6
β1
β2
β3
β4
β5
β6
y
10
Combine Reflections and Translations
Ex1: Write the equation for each function shown below.
Ex2: Graph the following functions.
π(π₯) =1
π₯ and π(π₯) = β
1
π₯+ 2 π(π₯) = βπ₯ and π(π₯) = βββπ₯
1.6 Assignment: Finish Ex1 & Ex2 above if needed. Then sketch each of the following functions. Include a
sketch of the parent function and check your answers using Desmos.
a) 3)( xxd b) 3)( xxf c) 25
1)(
xxh
d) 4)2()( xxj e) 1
1)(
xxk f) 31)( xxg
11
MCR3U 1.7 Stretches & Compressions of Functions Date: ____________
Stretches and compressions are transformations that cause functions to change shape.
1) Vertical Stretch or Compression :
* The graph of the function g(x) = af(x), a βΊ 0
β―β when |π| > 1, there is a VERTICAL STRETCH
by a factor of a.
β―β when 0 < |π| < 1, there is a VERTICAL COMPRESSION
by a factor of a.
Examples: Graph the following functions along with their parent functions.
a) π(π₯) = |π₯| and π(π₯) =1
2|π₯| b) π(π₯) = βπ₯ and π(π₯) = 3βπ₯
c) π(π₯) =1
π₯ and π(π₯) =
2
π₯
12
2) Horizontal Stretch or Compression :
* The graph of the function g(x) = f(kx), k βΊ 0
β when |k| > 1, there is a HORIZONTAL COMPRESSION
β― by a factor of 1
π
β when 0 < |k| < 1, there is a HORIZONTAL STRETCH
by a factor of 1
π
Examples : Graph the following functions along with their parent functions.
a) π(π₯) = βπ₯ and π(π₯) = β2π₯ b) π(π₯) = |π₯| and π(π₯) = |1
3π₯|
c) π(π₯) =1
π₯ and π(π₯) =
1
2π₯
13
Mapping Rule
A mapping rule looks at what happens to each point on the function when transformations are applied.
When multiple transformations are applied to a function using a mapping rule can be useful.
π(π₯) = π₯2 to π(π₯) = β(π₯ β 2)2 + 5
(π₯, π¦) β
π(π₯) = βπ₯ to π(π₯) = 2ββ(π₯ + 3) β 1
(π₯, π¦) β
π(π₯) = |π₯| to π(π₯) = β1
2|
1
3π₯| + 5
(π₯, π¦) β
If the point (3,8) is on the graph of π¦ = π(π₯) what is the image point on the graph of π¦ = β3π(2(π₯ β 1)) + 4
(π₯, π¦) β
1.7 Assignment: p.51 #1-3 p.59 #3, 4a, 5b, 6d, 7c p.70 #4 (ignore instructions, do mapping rule)
14
MCR3U 1.8 Combinations of Transformation Date: ____________
ββ―Stretches, compressions and reflections need to be performed before translations.
β β―The function must be written in the form y = af [k(x β d)]+ c to identify the specific β―transformations.
*Sometimes we need to factor out the k value to see the translation left or right.
Examples: Graph the following pairs of functions. Describe the transformations and state the mapping rule.
a) π(π₯) = βπ₯ and π(π₯) =3
2π(β(π₯ β 4)) β 1 Transformations:
Mapping Rule:
(x,y) ->
b) π(π₯) = |π₯| and π(π₯) = βπ (π₯
3) + 2 Transformations:
Mapping Rule:
(x,y) ->
15
c) π(π₯) =1
π₯ and π(π₯) = 2π(π₯ β 3) β 1 Transformations:
Mapping Rule:
(x,y) ->
d) π(π₯) = |π₯| and π(π₯) = β2π(0.5π₯ + 1) Transformations:
Mapping Rule:
(x,y) ->
d) π¦ = π(π₯) and π(π₯) = β1
3π(β4π₯ + 4) β 5 Transformations:
Mapping Rule:
(x,y) ->
1.8 Assignment: p.70 #6(ignore instructions, do mapping rule), 7bc, 8bc, 9bc, 10(ignore instructions, do
mapping rule), 11, 16