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Foundations & Pre-calculus 10
Chapter 5—Relations and Functions Outline
Date
Topic
5.1 Representing Relations
• table, set of ordered pairs, arrow diagram, etc..
5.3 Interpreting and Sketching Graphs
Assignment
p.262 #3a, 4,6,8
5.2 Properties of Functions Part I + 5.4 Graphing Data
• Review—practice—graphing tov
• What is a function
• Graphing data
• Domain and range given data (not graph)
• Discrete vs. continuous data
5.2 Properties of Functions Part II: Function Notation
• Function notation
5.5 Graphs of Relations and Functions
• Vertical line test for functions
• Number line Worksheet
• Domain and Range
o inequalities and interval notation
5.6 Properties of Linear Relations
• Detemine if it is linear from table, ordered pairs,
graph, equation (graph equation using table of
values)
• Determine rate of change from graph
5.7 Interpreting Graphs of Linear Functions
• Determine intercepts and domain/range
• Sketch graph given function notation—find t for
g(t)=-3
• Match graph to rate of change and y-intercept
Graphing Review worksheet
p.270 #1-5, 8,9
p.286 #1
Review
Chapter 5 Test
p. 281 #1-6,11,13,16,17
p.271 #6,7,14-19, 12
Function notation worksheet
Number line worksheet
p.293 #1-8,10-11,16,17,19
Domain and Range Worsheet
p.307 #1-5, 8,11,12,14,16,17
p. 319 #1-8,10,11, 15, 16, 17
p. 326 #3, 5, 6, 9-11, 13, 16, 18
Group Review
Foundations & Pre-calculus 10
5.1 Representing Relations
A set is ____________________________________________________________
An element is _______________________________________________________
A relation is ________________________________________________________
Relations can be represented in several ways:
As a table: The heading of each column describes each set.
Fruit
Apple
Apple
Blueberry
Cherry
Huckleberry
Colour
Red
Green
Blue
Red
Blue
As a set of ordered pairs: (element of first set, element of second set)
In words or a phrase:
An Arrow Diagram: The two ovals represent the sets. Each arrow associates an
element of the first set with an element of the second set.
When elements of one or both sets in the relation are numbers, the relation can
be represented as a ________________________________
Foundations & Pre-calculus 10
Ex. #1:Different breeds of dogs can be associated with their mean heights.
Consider the relation represented by this graph. Represent the relation as:
a) A table
b) An arrow diagram
Sometimes a relation contains so many ordered pairs that it is impossible to list all
of them or to represent them in a table.
Ex. #2:
a) Describe the relation in words.
b) List two ordered pairs that belong to the relation.
Foundations & Pre-calculus 10
5.3 Interpreting and Sketching Graphs
Discuss Make Connections and Construct Understanding.
Time
Speed
Speed
Speed
1. Tyler rides up and down a steep hill on his bike. Which graph best
represents his ride?
Time
Time
2. Draw a graph that is a reasonable representation of each of the following
situations. (Label your axes)
The temperature of water
in a pot on the stove
Your height over the
course of your lifetime
Your health over your
lifetime if you smoke
Define:
Dependent Variable: _________________________________________________
__________________________________________________________________
Independent Variable: ________________________________________________
__________________________________________________________________
Foundations & Pre-calculus 10
Ex. #1: Describe the journey for each segment of the graph.
Segment Graph
OA
The graph goes up to the right,
so as time increases, the
distance from Winnipeg
_________________.
AB
The graph is horizontal, so as
time increases, the distance
__________________________
BC
The graph goes up to the right,
so as time increases, the
distance __________________
CD
The graph is horizontal, so as
time increases, the distance
_________________________
DE
Journey
The graph goes down to the
right, so as time increases, the
distance __________________
What was the total driving time?
What are the dependent and the independent variables?
Foundations & Pre-calculus 10
Ex. #2: Karys went on a bicycle ride. She accelerated until she reached a speed of
20 km/h, then she cycled for 30 min. at approximately 20 km/h. Karys arrived at
the bottom of a hill, and her speed decreased to approximately 5 km/h for 10 min
as she cycled up the hill. She stopped at the top of the hill for 10 min.
Sketch a graph of speed as a function of time. Label each section of the graph,
and explain what it represents.
Foundations & Pre-calculus 10
Review—Practice –Graphing using a table of values and algebra
1. Find the value of y when 2
3 4
2. Find the value of x when 12
5 2
3. Find the value of x when 5
2 1
4. Find the value of y when 3
4 5
5. Graph the equation
6. Graph the equation
y = 2x + 1
x
-2
-1
0
1
2
y
6x – 2y = 8
x
y
Foundations & Pre-calculus 10
5.2 Properties of Functions Part I + 5.4 Graphing Data
A __________________ is a special type of relation where each element in the
first set is associated with exactly one element in the second set.
The set of first elements of a function is called the ______________________.
The set of related second elements of a function is called the _________________.
Ex. #1: Here are two different ways to relate vehicles and the number of wheels
each has.
This relation associates a number with a
vehicle with that number of wheels
This relation associates a vehicle with
the number of wheels it has
This diagram _____________________
a function because
This diagram _____________________
a function because
Display as a set of ordered Pairs:
Display as a set of ordered Pairs:
The set of ordered pairs _____________ The set of ordered pairs _____________
_________________ a function because a function because
Foundations & Pre-calculus 10
Ex. #2: Create a table showing the cost of movie tickets.
Variable
Number of
Tickets
Purchased
1
2
3
4
5
6
Total Cost
($)
Variable
a) Is this relation a function? Explain.
b) Identify the independent variable and the dependent variable. Explain.
c) Write the domain and range.
d) Graph the function. Do you connect the points? Explain.
Foundations & Pre-calculus 10
Function Notation
Function Notation:
y is a function of x means y depends on x for its value.
This can be expressed as f(x), said f of x or f at x. f is the name of the function and x is
the variable used in the function. When we give functions “names”, this is called
function notation.
When you see f(x) it means the same as the y value.
Example: The function y = 2 x − 1 can be expressed using function notation as
f(x) = 2x-1. Complete the chart below for each function.
Equation
Function
Notation
y = 2x – 1
y = 4x + 1
y = 5x – 7
y = x2 – 5x + 2
y = 2x
y = 3x2
y = 4 – 2x
Substitution:
If f(x) = 3x + 1, then this is the same as y = 3x + 1.
f(2), said f at 2, means replace x with 2 and find the value of y.
f(-3), said f at -3, means replace x with -3 and find the value of y.
Example:
f(x) = 3x + 1
f(2) = 3(2)+1
f(2) = 6+1
f(2) = 7
This means when x=2, y=7.
f(x) = 3x + 1
f(-3) = 3(-3) + 1
f(-3) = -9 + 1
f(-3) = -8
This means when x=-3, y=-8
Example:
Exercises:
Using the function, f(x) = 3x + 1 and g(x) = -4x – 4, find the following values:
f(7) =
g(-2) =
f(-4) =
g(0) =
Using the function, f(x) = x2 + 3x – 5, find the following values:
f(1) =
f(0) =
f(-2) =
f(4) =
Foundations & Pre-calculus 10
5.2 Properties of Functions Part II—Function Notation
Function Machine:
What is the rule for the function machine?
Which numbers would complete this table for the machine?
Input Output
1
5
2
7
9
4
13
Let x be the input value and y be the output value. Create an equation for this
function in terms of x and y.
Consider the following function machine—it calculates the value of quarters
When the input is q quarters, the output or
value, V, in dollars is:
The equation that describes this function is:
Since V is a function of q, we can write this
equation using function notation:
This notation shows that _____ is the dependent variable and that _____ depends
on _____
3 represents the value of the function when ___________
Foundations & Pre-calculus 10
Ex. #1: The equation 0.08 50 represents the volume, V litres, of gas
remaining in a vehicle’s tank after travelling d kilometres. The gas tank is not
refilled until it is empty.
a) Describe the function. Write the equation in function notation.
b) Determine the value of 600. What does this number represent?
c) Determine the value of d when 26. What does this number
represent?
Foundations & Pre-calculus 10
5.5 Graphs of Relations and Functions
The relation 2 is graphed:
We know that the relation 2 is a function because each
value of x associates with _______________________ value
of y, and each ordered pair has a different _____________________.
The __________________ of a function is the set of values of
the independent variable, or the set of ____________________
The __________________ of a function is the set of values of the dependent
variable, or the set of _______________________.
Vertical Line Test for a Function
A graph represents a function when no two points on the graph lie on the same
______________________________
Ex. #1: Which of these graphs represents a function?
Foundations & Pre-calculus 10
Ex. #2: Use the graph of the function 3 7
a) Determine the range value when the
domain value is 3.
b) Determine 1.
c) Determine the value of x when 1
d) Circle the coordinate that is represented by 1 10.
What are the coordinates at this point?
Ex. #3: Use the graph given of 4 3
a) Determine the domain value when the
range value is 5.
b) Determine 0).
c) Determine the value of x when 7
d) Circle the coordinate that is represented by 3 9.
What are the coordinates at this point?
Foundations & Pre-calculus 10
Domain and Range of a Graph
Visualize:
Domain—represented by the shadow
of its graph on the _______________
Range—represented by the shadow
of its graph on the _____________
You could also imagine ___________________ the function towards the axes.
Ex. #4: Determine the domain and range of the graphs of each function
Foundations & Pre-calculus 10
Interval Notation
A notation for representing an interval as a pair of numbers. The numbers are the
endpoints of the interval. Parentheses and/or brackets are used to show whether
the endpoints are excluded or included.
For example, 3, 8 is the interval of real numbers between 3 and 8, including 3
and excluding 8. Using inequalities, this interval is described as 3 8
Ex. #5: Determine the domain and range of the following relations, using words,
inequalities, and interval notation.
y
y
a)
b)
4
4
3
3
2
2
1
1
x
x
−4
−3
−2
−1
1
−1
2
3
4
5
−4
−3
−2
−1
1
−1
−2
−2
−3
−3
−4
−4
2
3
4
5
Foundations & Pre-calculus 10
5.6 Properties of Linear Relations
Discuss Make Connections on p.300
The cost for a car rental is $60, plus $20 for every 100km driven. So, the
independent variable is the _____________________________ and the
dependent variable is the _______________________.
We can identify that this is a linear relation in different ways.
• A table of values
Distance (km)
0
100
200
300
400
Cost($)
60
80
100
120
140
For a linear relation, a constant change in the independent variable results
in a ____________________________ in the dependent variable.
• A set of ordered pairs
0,60, 100,80, 200,100, 300,120, 400,140
• A graph
Foundations & Pre-calculus 10
We can use each representation to calculate the rate of change—the fraction:
The rate of change is _____________; that is, for each additional 1 km driven, the
rental cost increases by ___________. The rate of change is _________________
for a linear relation.
We can determine the rate of change from the equation that represents the
linear function.
Let the cost be C dollars and the distance driven by d kilometres.
An equation for this linear function is:
Ex. #1: Which relation is linear? Justify the answer.
a) A new car is purchased for $24000. Every year, the value of the car decreases
by 15%. The value is related to time.
Every year, the value decreases by 15%. The value of the car is:
_______________________________ of its previous value.
Time
Value ($)
(years)
0
1
2
3
Foundations & Pre-calculus 10
b) For a service call, an electrician charges $75 flat rate, plus $50 for each hour he
works. The total cost for service is related to time.
After the first hour, the cost increases by ______ per hour.
Time (h) Cost ($)
0
1
2
3
4
Examples of Linear Relations
Examples of Non-linear Relations
How can you tell from an equation whether the relation is linear?
Foundations & Pre-calculus 10
Ex. #3: A water tank on a farm near Swift Current, Saskatchewan, holds 6000L.
Graph A represents the tank being filled at a constant rate.
Graph B represents the tank being emptied at a constant rate.
a) Identify the independent and dependent variables.
Independent variable is ______________
Dependent variable is _______________
b) Determine the rate of change of each relation, then describe what it
represents.
Foundations & Pre-calculus 10
5.7 Interpreting Graphs of Linear Functions
Discuss Make Connections on p.311
Any graph of a line that is not vertical represents a function. We call these
functions ______________________________
Each graph below shows the temperature, T degrees Celsius, as a function of time,
t hours, for two locations
The point where the graph intersects
the horizontal axis is _____________
The horizontal intercept is ______
This point of intersection represents:
The point where the graph intersects
the vertical axis is _____________
The vertical intercept is ______
This point of intersection represents:
Domain:
Range:
The Rate of Change:
The point where the graph intersects
the horizontal axis is _____________
The horizontal intercept is ______
This point of intersection represents:
The point where the graph intersects
the vertical axis is _____________
The vertical intercept is ______
This point of intersection represents:
Domain:
Range:
The Rate of Change:
Foundations & Pre-calculus 10
We can use the intercepts to graph a linear function.
To determine the y-intercept, find the value of y when x=_______
That is evaluate when _________________________________
To determine the x-intercept, find the value of x when y=_______
That is determine the ______________________________________
Ex. #2: Sketch a graph of the linear function 2 7
Ex. #3: Which graph has a rate of change of ½ anda vertical intercept of 6?