Mathematics 1201 Unit 4: Relations and Functions (Chapter 5)

Sections 5.1: Representing Relations Set: a collection of distinct objects

Ex: {1, 2, 3, 4, 5} Element: one object in a set

Ex: 4 is an element in the above set ● The order of elements in the set does not matter

Relation: associates the elements of one set with the elements of another set

● A relation can be represented as ordered pairs, a table or as an arrow diagram Example 1: Representing a Relation Given as a Table Northern communities can be associated with the territories they are in. Consider the relation represented by this table.

a) Describe this relation in words.

b) Represent this relation: i) as a set of ordered pairs ii) as an arrow diagram

Example 2: Representing a Relation Given as a Bar Graph Different breeds of dogs can be associated with their mean heights. Consider the relation represented by this graph. Represent the relation:

a) as a table

b) as an arrow diagram

Example 3: Identifying a Relation from a Diagram In this diagram:

a) Describe the relation in words

b) List 2 ordered pairs that belong to the relation

Complete the following questions on p. 262-263: #3-8, 10

Section 5.2: Properties of Functions Domain: the set of first elements of a relation Range: the set of related second elements of a relation Function: a special type of relation where each element in the domain is associated

with exactly one element in the range Example 1: Identifying Functions For each relation below:

● Determine whether the relation is a function. Justify the answer ● Identify the domain and range of each relation that is a function

a) A relation that associates given shapes with the number or right angles in the shape:

{ (right triangle, 1), (acute triangle, 0), (square, 4), (rectangle, 4), (regular hexagon, 0) }

Function?

Domain:

Range:

b) Function?

Domain:

Range:

Example 2: Describing Functions 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 your choices.

c) Write the domain and range.

Note: ● We can think of a function as an input/output machine. The input can be any number

in the domain, and the output depends on the input number. So, the input is the independent variable and the output is the dependent variable.

● Any function that can be written as an equation in two variables can be written in

function notation, and vice versa.

○ To write the equation d = 4t + 5 in function notation, we may write d(t) = 4t + 5, where t represents an element of the domain and d(t) represents an element of the range.

Example 3: Using Function Notation to Determine Values 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?

Complete the following questions on p. 270-273: #4, 5, 8, 9, 10, 12, 14, 15, 18, 19

Section 5.3: Interpreting and Sketching Graphs The properties of a graph can provide information about a given situation:

Example 1: Interpreting a Graph 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 2: Describing a Possible Situation for a Graph Describe the journey for each segment of the graph.

Segment Graph Journey

OA

AB

BC

CD

DE

Example 3: Sketching a Graph for a Given Situation 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. Use graph paper to sketch a graph of speed as a function of time. Label each section of the graph, and explain what it represents.

Segment Journey

OA

AB

BC

CD

DE

EF

Complete the following questions on p. 281-283: #3-11, 13

Section 5.4: Graphing Data (Math Lab) Section 5.5: Graphs of Relations and Functions Domain: the domain of a function is the set of values of the independent variable

(x-values)

Range: the range of a function is the set of values of the dependent variable (y-values)

Vertical Line Test for a Function A graph represents a function when no two points on the graph lie on the same vertical line. Place a ruler vertically on a graph, then slide the ruler across the graph. If one edge of the ruler always intersects the graph at no more than one point, the graph represents a function.

Example 1: Identifying whether a Graph represents a Function Which of these graphs represents a function? Justify your answer.

Example 2: Determining the Domain and Range of the Graph of a Function (Worksheet) Determine the domain and range of the graph of each function.

Example 3: Determining the Domain and Range of the Graph of a Situation This graph shows the number of fishing boats, n, anchored in an inlet in the Queen Charlotte Islands as a function of time, t.

a) Identify the dependent variable and the independent variable.

b) Why are the points on the graph NOT connected?

c) Determine the domain and range of the graph.

Example 4: Determining the Domain Values and Range Values from the Graph of a Function 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.

Complete the following questions on p. 294-297: #4-11, 16, 17

Section 5.6: Properties of Linear Relations Example 1: Determining whether a Table of Values Represents a Linear Relation Which of the table of values represents a linear relation? Justify your answer.

Example 2: Determining whether an Equation Represents a Linear Relation

a) Using graph paper, graph each equation: i) y = -3x + 25

ii) y = 2x2 + 5 iii) y = 5 iv) x = 1

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

Example 3: Identifying a Linear Relation Which relation is linear? Justify your answer.

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.

Time (years) Value ($)

b) For a service call, an electrician charges a $75 flat rate, plus $50 for each hour he works. The total cost for service is related to time.

Time (h) Cost ($)

Example 4: Determining the Rate of Change of a Linear Relation from Its Graph A water tank on a farm near Swift Current, Saskatchewan, holds 6000 L. 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.

b) Determine the rate of change of each relation, then describe what it represents.

Complete the following questions on p. 308-310: #3-7, 10, 12, 14, 15, 16, 17

Section 5.7: Interpreting Graphs of Linear Functions Horizontal Intercept: the point where the graph intersects the horizontal axis

(x-intercept). Written as a coordinate point (#, 0) Vertical Intercept: the point where the graph intersects the vertical axis

(y-intercept). Written as a coordinate point (0, #)

Example 1: Determining Intercepts, Domain, and Range of the Graph of a Linear Function 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 vertical and horizontal intercepts. Describe what the points of intersection represent.

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

Example 2: Sketching a Graph of a Linear Function in Function Notation Using graph paper, sketch a graph of the linear function f(x) = -2x + 7. Workings:

Example 3: Matching a Graph to a Given Rate of Change and Vertical Intercepts Which graph has a rate of change of and a vertical intercept of 6? Justify your answer.2

1

Example 4: Solving a Problem Involving a Linear Function This 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?

Complete the following questions on p. 319-323: #4-11