1. No category

1
Math 10 C (AP)
Chapter 5: Relations & Functions
Page 254
Review:
–3
What is 7 2 expressed in radical form?
Lesson 5.1
Representing Relations
Page 256
Key Terms
Relation – a rule that associates the elements of one set with the elements of another set (a set of ordered pairs).
Set – a collection of distinct objects
Element – One distinct object in a set.
For example {2, 4, 6, 8} = a set of natural even numbers with elements 2, 4, 6, and 8.
When we represent relations using numbers, a relation is a set of ordered pairs. The elements in the relation are
the numbers that represent specific coordinate points on a Cartesian plane.
For example {(2,4), (4,8), (6,12)} is a relation.
Relations can be represented in a number of ways:
1. Table of Values
2. Graphs
3. Arrow Diagrams (mapping diagram)
4. Equations
5. In Words (description)
Example #1 – Represent the following relation in 5 different ways:
{(1,3), (2,4), (3,5)}
Table of Values
x
1
2
3
y
3
4
5
Recall: (x, y)
Graph
Y
6
Equation
Words
𝑦 =𝑥+2
y is always equal to
the value of x plus
two.
5
4
3
2
1
X
0
1
2
3
4
5
6
Created with a trial version of Advanced Grapher - http:/ / www.alentum.com/ agrapher/
Explain why we wouldn’t connect the points in the graph!
2
Math 10 C (AP)
Ex #2 Draw an arrow diagram for the following relation:
{(3,7), (0,4), (-1,0), (-5,-1)}
x
y
-
Ex #3 Represent the following arrow diagram in the form of an equation:
x
y
-3
0
-2
-1
0
1
1
4
2
3
9
Ex. #4 Animals can be associated with the classes they are in.
Animal
Ant
Eagle
Snake
Turtle
Whale
a. Describe this relation in words.
b. Represent this relation:
i) as a set of ordered pairs
ii) as an arrow diagram:
Class
Insect
Aves
Reptilian
Reptilian
Mammalia
3
Math 10 C (AP)
When the elements of either one or both sets in a relation are numbers, the relation can be represented as a bar
graph as well (not always line graphs).
Ex. #5
Different towns in British Columbia can be associated with the average time, in hours, it takes to drive to
Vancouver. Consider the relation represented by the following graph.
Average Travel Time to Vancouver
5
4
Time
(h)
3
2
1
0
Horeshoe Bay
Lilloet
Pemberton Squamish
Whistler
Town
a) As a table
Town
Horseshoe Bay
Lilloet
Pemberton
Squamish
Whistler
Average Time(h)
0.75
4.5
2.75
1.5
2.5
b) as an arrow diagram:
Horseshoe Bay
2.75
Lilloet
2.5
Pemberton
4.5
Squamish
0.75
Whistler
1.5
4
Math 10 C (AP)
Relation vs. Function
Recall: A relation is a set of ordered pairs. A relation produces one or more output numbers for every valid
input number.
Ex. {(-2, 3), (3, 4), (4, 7), (-2, 6)}
Notice that we get 2 different output numbers for an input of -2. (3 and 6)
A function is also a set of ordered pairs, however, for every valid input number, there is only ONE output
number.
Ex. {(0, 2), (2, 6), (3, 8), (4, 10)
In order to determine if a graph is a relation that is NOT a function, or whether it is in fact a function, we can
use what we call the “Vertical Line Test”. This means that if we draw a vertical line anywhere on the graph,
and 2 or more points on the graph are touching (lie on) that line, then the graph is NOT a function.
Ex. 6
Which of the graphs below are relations? Which are functions?
10
Y
Y
10
5
5
X
X
-10
-5
0
5
-10
10
0
-5
-10
-10
Created with a trial version of Advanced Grapher - http:/ / www.alentum.com/ agrapher/
Created with a trial version of Advanced Grapher - http:/ / www.alentum.com/ agrapher/
Y
10
X
X
-10
-5
Y
5
5
0
10
-5
-5
10
5
5
10
-10
-5
0
5
10
-5
-5
-10
-10
Created with a trial version of Advanced Grapher - http:/ / www.alentum.com/ agrapher/
Created with a trial version of Advanced Grapher - http:/ / www.alentum.com/ agrapher/
Assignment: Pg. 262-263 #3-8, 10
and
Pg.
5
Math 10 C (AP)
Lesson 5.2
Review
Properties of Functions
Page 264
Simplify each expression. Express the answer using positive exponents.
a)
 32   13  
 5   5  
    
6
b)
21 a 2 b 3
4a 3b 5
Key Terms:
Domain ~ The set of first elements in a relation. (Independent Variable)
Range ~ The set of second elements in a relation. (Dependent Variable)
Function ~ A special type of relation where each element is the domain is associated with exactly one element
in the range.
Example #1 ~ A FUNCTION: Cost of Cereal in a Grocery Store
State the domain and range of this function.
Domain: {
Range: {
{(𝐶ℎ𝑒𝑒𝑟𝑖𝑜𝑠, $3.99)(𝐽𝑢𝑠𝑡 𝑅𝑖𝑔ℎ𝑡, $5.99)(𝑀𝑖𝑛𝑖 − 𝑊ℎ𝑒𝑎𝑡𝑠, $4.99)(𝑆𝑝𝑒𝑐𝑖𝑎𝑙 𝐾, $5.99)}
6
Math 10 C (AP)
Example #2 ~ A FUNCTION: Grade ten student and Their Age
State the domain and range of this function.
Domain: {
Range: {
Write the associated ordered pairs for this function.
Example #3
NOT A FUNCTION
State the domain and range of this relation.
Domain: {
Range: {
Write the associated ordered pairs for this relation.
Example #4 ~ Function Notation
The equation 𝐶 = 25𝑛 + 1000 represents the cost, C, dollars, for a feast following a sports competition, where
n is the number of people attending.
a) Describe the function. Write the equation in function notation.
b) Determine the value of 𝐶(100). What does this number represent?
c) Determine the value of n when 𝐶(𝑛) = 5000. What does this number represent?
d) Write 𝐶(10) = 1250 as an ordered pair.
Assignment: Pg. 270 4 to 6, 8 to 10, 12 to 19
7
Math 10 C (AP)
Lesson 5.3
Interpreting and Sketching Graphs
Page 264
Many times when interpreting graphs, we imagine that our x and y axes represent some specific everyday
variables. These variables are known as dependent and independent variables.
Dependent Variable
 the variable whose value is dependent upon or determined by another variable.
 the variable that is ALWAYS on the y-axis!
** Common dependent variables include distance, speed, height, and temperature.
Independent Variable
 in a relation, the variable whose value may be freely chosen and upon which the value(s) of the other
variable depend.
 the variable that is ALWAYS on the x-axis!
** The most common independent variable used in mathematics and science is TIME.
The properties of a graph can provide information about a given situation.
Steep decrease
Moderate Increase
Distance
(m)
Steep decrease
No change
Moderate Decrease
Steep increase
Time (min)
http://graphingstories.com/
Graphing Stories:
https://www.youtube.com/watch?v=ReuackmLI9I&list=PL9_QzgM9GGBjUgmS-m-9e5uWLowmdJnPE
8
Math 10 C (AP)
Graphing Story #1
Tell me about it.
Graphing Story #5
Tell me about it.
9
Math 10 C (AP)
Graphing Story #6
Tell me about it.
Example #1
What do you think a Bungee jump would look like as a Height of Head vs. Time Graph? Justify your answer
with an explanation.
Example #2 ~ Match each scenario with its appropriate graph.
a) the vertical growth of a tree
b) the temperature of a cup of hot chocolate as it sits on a table
c) Calgary’s daily mean temperature
10
Math 10 C (AP)
Example #3
Sailia leaves home and travels at a steady speed of 100 km/h for 1 h, slows to 80 km/h for
1
2
h due to
construction, then continues at 110 km/h for another hour. She stops for 2 h for a meeting. She returns home at a
steady 100 km/h.
Sketch a graph of each scenario.
a)
Distance Travelled vs. Time
b) Distance from Home vs. Time
c)
Speed vs. Time
11
Math 10 C (AP)
Example #4
Draw a graph that coordinates with a person going for a jog. Use the following information to help create your
graph:
This is the path of a person out for a jog:
 1st 20 minutes: getting ready for the jog and then stretching
 2nd 20: jogs away from the house for 2.0 km
 next 10-15 minutes: person starts to walk.
 Next 10 minutes: when 2.5 km from home, they stop to rest.
 Next 15 minutes: starts to jog home
 Next 10 minutes: When 1.5km from home, the jogger stops again.
Assignment: pg. 281-283 #1-11, 13, 17
and pg. 286 1a and 2a
12
Math 10 C (AP)
Lesson 5.5
Graphs of Relations and Functions
Review: Factor completely.
a) 16x2 – 4y2
(x – 2)2
b) y4 – 10y2 + 25
Page 287
c) 10y2 + 9y + 2
d) (x + 2)2 –
Chapter 5 Review…
Vertical Line Test for a Function:
 A graph represents a function when no two points on the graph lie on the same vertical line.
Domain
 The set of the first elements in a certain number of specific ordered pairs of a relation
** Domain ALWAYS refers to the x coordinates of ordered pairs**
Range
 The set of the second elements in a certain number of specific ordered pairs of a relation.
** Range ALWAYS refers to the y coordinates of ordered pairs**
Example #1
For each relation below:
 Determine whether the relation is a function.
 Identify the domain and range of each relations that is a function
a) {(7, 2), (17, 2), (18, 3), (21, 2), (17, 3)}
b)
13
Math 10 C (AP)
Graphing
A scatterplot is the result plotting data on a Cartesian (coordinate) plane that can be represented as ordered
pairs.
When graphing, if the domain and range are only “elements” of the graph, we DO NOT connect the points on
the graph. If the domain or range is all the real numbers (𝑅), then we connect the points and draw arrows at
both ends of the line.
For example: {(2,5), (4,7), (5,9), (9,12)} --- We would NOT connect the points
y = 3x + 1 --- We would connect the points (since x could represent any Real number)
Example #2
Graph the equation y = 3x + 1 for the domain {-1, 0, 1, 2, 3}, and for the domain of the Real numbers (R).
** Need to set up a Table of Values**
x
3x+1
D: {-1, 0, 1, 2, 3}
y
(x, y)
D: Real Numbers
14
Math 10 C (AP)
Example #3 ~ You Try
Graph the equation y = x + 4 for the domain {-3, -2, -1, 0, 1}, and for the domain of the Real numbers (R).
x
x+4
y
(x, y)
D: {-3, -2, -1, 0, 1}
D: Real Numbers
Discrete vs. Continuous
Discrete Graph
 A graph that is a series of separate points
 The points are NOT connected on the graph
Continuous Graph
 A line graph in which the line in unbroken
 The points on the graph ARE connected
In the above example, the first graph is discrete (D: {-3, -2, -1, 0, 1}) and the second graph is continuous (D:
Real Numbers).
For example: “If I leave here at Noon and drive to Calgary, I arrive at 3:00pm. Where was I at 1:30pm? Did I
get straight to Red Deer? No.” I was driving for the duration of time in between, therefore this is continuous
(time is continuous) because at any specific time I was at a specific point on the journey.
“If I count the number of students I talk to during the day…I talk to one…then two…etc. Can I talk to 4.23
students? No.” There is no in between…therefore, this is discrete.
15
Math 10 C (AP)
Example #4
Are the following graphs discrete or continuous?
10
Y
10
5
Y
5
5
X
-10
0
-5
5
10
Y
10
X
-10
-5
0
5
X
10
-10
-5
0
-5
-5
-5
-10
-10
-10
5
10
Created with a trial version of Advanced Grapher - http:/ / www.alentum.com/
Created with agrapher/
a trial version of Advanced Grapher - http:/ / www.alentum.com/
Created with aagrapher/
trial version of Advanced Grapher - http:/ / www.alentum.com/ agrapher/
Example #5
Determine the domain and range of the following graphs:
a.)
b)
Y
10
Y
10
5
5
X
-10
-5
0
5
X
10
-10
-5
-5
-10
-10
Created with a trial version of Advanced Grapher - http:/ / www.alentum.com/ agrapher/
c)
0
-5
10
Y
5
Created with a trial version of Advanced Grapher - http:/ / www.alentum.com/ agrapher/
d)
10
5
Y
5
X
-10
-5
0
10
5
10
-5
X
-10
-5
0
5
10
-5
-10
-10
Created with a trial version of Advanced Grapher - http:/ / www.alentum.com/ agrapher/
Created with a trial version of Advanced Grapher - http:/ / www.alentum.com/ agrapher/
d) Show an example where the domain could be −2 ≤ 𝑥 ≤ 6, and the range 0 ≤ 𝑦 ≤ 4.
Assignment: Pg. 293-297 #1-11, 14, 16, 17
16
Math 10 C (AP)
Lesson 5.6 ~ Properties of Linear Relations
Review
When 80 is simplified using the form a b , what is the numerical value of the radicand?
Page 300
Function notation is a short form version of representing an equation.
For example:
In x and y notation:
𝑦 = 3𝑥 + 4
In function notation: 𝑓(𝑥) = 3𝑥 + 4
“f of x”
In function notation f names a function. We say “f of x” because it is the value of the function at point x = ___.
In function notation, we use substitution to solve the values of the function given a particular x value.
Example #1
If 𝑓(𝑥) = 3𝑥 + 4, find f(3), f(0), and f(-10).
Example #2
If 𝑔(𝑥) = 3𝑥 2 − 2, find:
g(2)
The value of x if g(x) = 25
g(-4)
17
Math 10 C (AP)
In order to graph linear equations, all we need to do is find 2 points on the graph and connect them with a
straight edge. There are various methods we can use to determine 2 points on a graph. We will use the various
methods in the following examples.
Example #1
Graph 𝑦 = 3𝑥 − 4, x 
Table of Values
ordered pairs
x
y
(x, y)
f(x)
Since x  , then we must either draw arrows on the ends of ours graphs, or make sure the line (graph)
is drawn to the end of grid that you are graphing on.
x – intercept(s)
 This is the point where the function or relation crosses the x-axis.
 Can also be referred to as the horizontal intercept
 y is always equal to zero. 𝒚 = 𝟎
y – intercept(s)
 This is the point where the function or relation crosses the y-axis.
 Can also be referred to as the vertical intercept.
 x is always equal to zero. 𝒙 = 𝟎
You can graph a function using just the x and y intercepts.
In order to find the x and y intercepts of a linear equation, simply let the opposite variable equal zero (0) in the
equation, and solve for the remaining variable.
18
Math 10 C (AP)
Example #2
Graph 𝑓(𝑥) = −2𝑥 + 6 for 𝑥 ∈ 𝑅. Use the x and y intercepts.
Example #3 ~ You Try
Graph 𝑓(𝑥) = −2𝑥 − 5, 𝑥 ∈ 𝑅
and
𝑦 = 2𝑥 + 3, 𝑥 ∈ 𝑅
Example #4
Graph each equation using a table of values. Which equations represent linear functions?
𝑥 = −2
𝑦 =𝑥+3
𝑦=4
𝑦 = 𝑥2 − 3
19
Math 10 C (AP)
Rate of Change
 The change in one quantity with respect to the change in another quantity.
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝐷𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒
 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝐼𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠
Example #5
A hot tub contains 1600 L of water. Graph A represents the hot tub being filled at a constant rate. Graph B
represents the hot tub being emptied at a constant rate.
a) Identify the dependent and independent variables.
b) Determine the rate of change of each relation, then describe what it represents.
Example #6
The table of values shows the cost of a pizza with up to 5 extra toppings.
Number
of Extra
Toppings
0
1
2
3
4
5
Cost
($)
12.00
12.75
13.50
14.25
15.00
15.75

Graph the data labeling the
independent and dependent
variables.

What patterns do
you see in the
function?

Can you represent
it as a linear
function?
20
Math 10 C (AP)
Example #7
The cost for a car rental is $60, plus $20 for every 100km driven. The independent variable is the distance
driven and the dependent variable is the cost. We can identify that this is a linear relation in different ways.

Create a Table of Values:
Distance (km)

Cost ($)
Graph the data labeling the independent and dependent
variables.
For a linear relation, a constant change in the independent variable results in a constant change in the dependent
variable.
We can see these patterns using table of values, ordered pairs or graphs.
What characteristics can you see in the graph of a linear relation?
We can use each representation to calculate the rate of change. The rate of change can be expressed as a
fraction:
𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒅𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒕 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆
$𝟐𝟎
=
= $𝟎. 𝟐𝟎/𝒌𝒎
𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒊𝒏𝒅𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒕 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆 𝟏𝟎𝟎𝒌𝒎
The rate of change is $0.20/km. For each additional 1km driven, the rental cost increases by 20 cents. The rate
of change is constant for a linear relation.
We can determine the rate of change and y-intercept for a linear function and from those, we can find the
equation that represents the function.
𝑦=
Write an equation for this linear relation:
𝑥+
21
Math 10 C (AP)
Example #8
Consider each relation. Determine whether the relation is linear. Explain why or why not.
a.
x
-9
-7
-5
-3
-1
y
-10
-5
0
5
10

Is there a constant rate of change
for both the independent and
dependent variables?

When the ordered pairs are
plotted, would the relation be a
linear function?
b. The graph shows the relationship between the amount, A, of a radioactive isotope present and the age
of a rock sample over time, t, years.
Example #9
Which relation is linear? Justify your answer.
a) A dogsled moves at an average speed of 10km/h along a frozen river. The distance travelled is related to
time.
b) The area of a square is related to the side length of the square.
Assignment: pg. 307-310 #3-10, 12-19
22
Math 10 C (AP)
Lesson 5.7 ~ Interpreting Graphs of Linear Functions
Review: Determine the Domain and Range. Which of the relations below are functions?
Page 311
23
Math 10 C (AP)
Example #1 ~ Page 311 Textbook example
Float planes fly into remote lakes in Canada’s Northern wilderness areas for ecotourism. This graph shows the
height of a float plane above a lake as the plane descends to land.
Where does the graph intersect the vertical axis? What does this point represent?
Where does the graph intersect the horizontal axis? What does this point represent?
What is the rate of change for this graph? What does it represent?
What would the domain and range of this graph?
24
Math 10 C (AP)
Example #2
This graph shows how the height of a burning candle changes with time.
a. Write the coordinates of the points where the graph intersects the axes. Determine the vertical and horizontal
intercepts. Describe what the points of intersection represent.
b. What are the domain and range of this function?
We can use the intercepts to graph a linear function written in function notation.
To determine the y-intercept, evaluate f(x) when x = 0; that is evaluate f(0).
To determine the x-intercept, determine the value of x when f(x) = 0.
Example #3
Sketch a graph of the linear function f(x) = 4x – 3 by finding the intercepts.
25
Math 10 C (AP)
Example #4
Which graph has a rate of change of -5 and a vertical intercept of 100? Justify your answer.
Example #5
This graph shows the total cost for a house call by an electrician for up to 6h work.
a) The electrician charges $190 to complete a job. For how many hours did she
work?
b) If she charges $50 for the first hour $30 for each additional hour rounding to the nearest hour, how
would this function be graphed?
Assignment: pg. 319-323 #1-12, 14-16, 18, 19
Read Pages 324 – 326
Assignment: Pages 326 to 328
26
Math 10 C (AP)
Domain and Range in Interval Notation:
Interval Notation:
(description)
Open Interval: (a, b) is interpreted as a < x < b where the
endpoints are NOT included.
(diagram)
(1, 5)
(While this notation resembles an ordered pair, in this context it
refers to the interval upon which you are working.)
Closed Interval: [a, b] is interpreted as a < x < b where
the endpoints are included.
[1, 5]
Half–Open Interval: (a, b] is interpreted as a < x <
b where a is not included, but b is included.
(1, 5]
Half–Open Interval: [a, b) is interpreted as a < x < b where
a is included, but b is not included.
[1, 5)
Non–ending Interval:
is interpreted as x > a where a
is not included and infinity is always expressed as being
"open" (not included).
Non–ending Interval:
is interpreted as x < b where
b is included and again, infinity is always expressed as
being "open" (not included).
We could even show no limits by using this notation: (–∞, +∞)
27
Math 10 C (AP)
Extra Practice: Find the domain and range for each graph in both set and interval notation.
Test Preparation: Practice Test on Page 329 ~ complete all questions