chapter 5 relations and functions. chapter 5 5.1 – representing relations

Chapter 5RELATIONS AND

FUNCTIONS

Chapter 55.1 – REPRESENTING

RELATIONS

NEW TERMS

A set is a collection of distinct objects.

An element of a set is one object in the set.

A relation associates the elements of one set with the elements of another.

EXAMPLE

Communities can be associated with the territories they are in. Consider the relation represented in this table.a) Describe this relation in

words.b) Represent this relation:

i) As a set of ordered pairs

ii) As an arrow diagram.

a) The relation shows the association “is located in” from a set of northern communities to a set of territories. For example, Yellowknife is located in the Northwest Territories.

b) If we listed them as ordered pairs it would look like this:

{(Hay River, NWT), (Iqaluit, Nunavut), (Nanisivik, Nunavut), (Old Crow, Yukon), (Whitehorse, Yukon), (Yellowknife, NWT)}

Different breeds of dogs can be associated with their mean heights. Consider this relation. Represent it as a table, and as an arrow diagram.

EXAMPLE

EXAMPLE

In this diagram:a) Describe the relation in words.b) List 2 ordered pairs that belong in the relation.

Independent Practice

PG. 262-263, #5, 6, 7, 9, 11, 12, 14.

Chapter 55.2 – PROPERTIES OF

FUNCTIONS

INPUT/OUTPUT

We can think of relations as similar to this input/output machine. We put a number in, and do a few things to it, and then get the output.

x is the input, and y is the output.

What is the rule for the input/output table below?

DOMAIN AND RANGE

The set of first elements of a relation is called the domain.

The set of related second elements of a relation is called the range.

A function is a special type of relation where each element in the domain is associated with exactly one element in the range.

FUNCTIONS

A function is a special type of relation where each element in the domain is associated with exactly one element in the range.

Which of these is a function?

FUNCTIONS

List a set of ordered pairs: List a set of ordered pairs:

Domain and Range: Domain and Range:

EXAMPLE

For each relation below:• Determine whether the relation is a function. • Identify the domain and range of each relation

that is a function.a) A relation that associates given b)

shapes with the number of right angles in the shape: {(right triangle, 1), (acute triangle, 0), (square, 4), (rectangle, 4), (regular hexagon, 0)}

INDEPENDENT & DEPENDENT VARIABLES

In the workplace, a person’s pay, P dollars, often depends on the number of hours worked, h. So, we say that P is the dependent variable. Since the number of hours work, h, does not depend on the pay, we say that h is the independent variable.

EXAMPLE

The table shows the masses, m grams, of different numbers of identical marbles, n.a) Why is this relation also a function?b) Identify the independent variable and the dependent variable. Justify

the choices.c) Write the domain and range.

Independent practice

PG. 270-273, #4, 5, 8, 9, 11, 12.

EXAMPLE

The equation V = –0.08d + 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 V(600) What does this number represent?c) Determine the value of d when V(d) = 26. What does this number

represent?

Function notation is a different way to write an equation for a relation. It just gives us a way to identify the dependent and independent variables in the equation itself.

a) So, in this equation, which is the independent variable? Why?

The amount of gas remaining depends on how far you’ve travelled—it doesn’t make sense the other way around.

Instead of V, we use V(d), which means V depending on d.

V(d) = –0.08d + 50

EXAMPLE

The equation V = –0.08d + 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 V(600) What does this number represent?c) Determine the value of d when V(d) = 26. What does this number

represent?V(d) = –0.08d + 50

b) When it says V(600) that means that we want to evaluate the expression for when d = 6.

V(600) = –0.08(600) + 50V(600) = 2

That means that after travelling 600 km, there will be 2 litres of gas left.

c) We let V(d) = 26

26 = –0.08d + 50

–24 = –0.08d

300 = d

When you have driven for 300 km there will be 26 L of gas left.

Independent practice

PG. 271-273, #6, 14, 15, 17, 18, 22.

Chapter 5

5.3 – INTERPRETING AND SKETCHING

GRAPHS

GRAPHS

EXAMPLE

Each point on this graph represents a bag of popping corn. Explain the answer to each question below.a) Which bag is the most expensive? What does it cost?b) Which bag has the least mass? What is this mass?c) Which bags have the same mass? What is this mass?d) Which bags cost the same? What is this cost?e) Which of bags C or D has the better value for money?

EXAMPLE

Describe the journey for each segment of the graph.

EXAMPLE

Samuel went on a bicycle ride. He accelerated until he reached a speed of 20 km/h, then he cycled for 30 min at approximately 20 km/h. Samuel arrived at the bottom of a hill, and his speed decreased to approximately 5 km/h for 10 min as he cycled up the hill. He stopped at the top of the hill for 10 min. Sketch a graph of speed a function of time. Label each section of the graph, and explain what it represents.

Independent practice

PG. 281-283, #3, 5, 7, 8, 9, 12, 14

Chapter 55.4 – GRAPHING

DATA

GRAPHING

To rent a car for less than one week from Ace Car Rentals, the cost is $65 per day for the first three days, then $60 a day for each additional day.

• Why are the points on the graph not joined?

• Is this relation a function? How can you tell?

• What is the domain? What is the range?

Independent practicePG. 286, #1, 2

You need to complete this on graph paper, and hand it in. Your graphs need to be neat and labeled.

Chapter 5

5.5 – GRAPHS OF RELATIONS AND

FUNCTIONS

RELATIONS AND FUNCTIONS

In an environmental study, Joe collected data on the numbers of different species of bird he heard or saw in a 1-h period every 2 h for 24 h. Alice collected data on the temperature in the area at the end of each 1-h period. They plotted their data.

Does each graph represent a relation? How about a

function?

Which of these graphs should have the data points

connected?

VERTICAL LINE TEST

A graph represents a function when no two points on the graph lie on the same vertical line.

TRY IT! ARE THESE FUNCTIONS?

EXAMPLE

Determine the domain and range of the graph of each function.a) b)

EXAMPLE

This graph shows the number of fishing boats, n, anchored in an inlet in the Queen Charlotte Islands as a function time, t.a) Identify the dependent variable and the independent

variable. Justify.b) Why are the points on the graph not connected? c) Determine the domain and range of the graph.

EXAMPLE

Here is a graph of the function f(x) = –3x + 7.a) Determine the range value when the domain value is –2.b) Determine the domain value when the range value is 4.

Independent practice

PG. 294-297, #4, 7, 8, 9, 11, 14, 16, 17

Chapter 55.6 – PROPERTIES OF

LINEAR RELATIONS

LINEAR RELATIONS

The cost for a car rental is $60, plus $20 for every 100 km driven. The independent variable is the distance driven and the dependent variable is the cost.

Table of Values:

For a linear relation, a constant change in the independent variable results in a constant change in the dependent variable.

LINEAR RELATIONS

As a set of ordered pairs:

A graph:

RATE OF CHANGE

Rate of change (also known as slope) tells you the steepness of a graph. It is the rate at which the dependent variable is changing at.

You can pick any segment of the graph to look at—if it’s a linear graph, it will always give you the same amount. Rise is the vertical distance between two points, while run is the horizontal distance between them.

LINEAR EQUATIONS

General form of a linear equation: y = mx + b

dependent variable

slope (or rate of change)

independent variable

y-intercept (or initial amount)

slope = 0.2y-intercept = 60

C = 0.2d + 60

EXAMPLE

Which table represents a linear equation?

EXAMPLE

a) Graph each equation.i) y = –3x + 25 ii) y = 2x2 + 5iii) y = 5 iv) x = 1

b) Which equations in part a represent linear relations? How do you know?

EXAMPLE

Which relation is linear? Justify.a) A new car is purchased for $24 000. Every year, the value of the car

decreases by 15%. The value is related to time.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.

EXAMPLE

A water tank on a farm near Swift Current, Saskatchewan holds 6000 L. Graph A represents the tank being filled at a constant rate, while Graph B represents the tank being emptied at a constant rate.a) Identify the independent and dependent variables.b) Determine the rate of change of each relation, then describe what it

represents.

Independent practice

PG. 308-310, #3, 5, 6, 7, 10, 12, 14, 17,

19.

Chapter 5

5.7 – INTERPRETING GRAPHS OF LINEAR

FUNCTIONS

LINEAR FUNCTIONS

The point where the graph intersects the horizontal axis (or the x-axis) is called either the horizontal incercept or the x-intercept.

The point where the graph intersects the vertical axis (or the y-axis) is called either the vertical incercept or the y-intercept.

What’s the domain and range for this graph?

What is the slope?

EXAMPLE

This graph shows the fuel consumption of a scooter with a full tank of gas at the beginning of a journey.a) Write the coordinates of the points where the graph intersects the axes.

Determine the x and y-intercepts. Describe what the points of intersection represent.

b) What are the domain and range of this function?

EXAMPLE

Sketch a graph of the linear function f(x) = –2x + 7.

EXAMPLE

Which graph has a rate of change of ½ and a vertical intercept of 6? Justify.

EXAMPLE

The graph shows the cost of publishing a school yearbook for College Louis-Riel in Winnipeg. The budget for publishing costs is $4200. What is the maximum number of books that can be printed?

Independent practice

PG. 319-323, #5, 6, 8, 9, 13, 14, 15, 17,

19