Math 10C

Notes Chapter 5

RED

Section 5.1 Representing Relations Math 10C

New words!!!!

A SET is a collection of distinct objects. An ELEMENT is one particular object inside a set. A RELATION associates the elements of one set with the elements of another

set.

Example:

Set 1: Types of Fruit Set 2: Colours of Fruit

An Apple

May have the colour

Element of Association Element of Set 1 Set

A possible set of ordered pairs for this relation would be:

{(apple, red), (apple, green), (blueberry, blue), (cherry, red), (huckleberry, blue)}

Can you think of 3 more potential elements that would fulfill the relationship of this set?

1. ______________________

2. ______________________

3. ______________________

There are a couple of other ways to illustrate a relation.

Chart Arrow Diagram

The order of the words in the ordered pairs is VERY important. As such, a relation has direction from one set to another set.

You Try:

Northern communities can be associated with the territories they are in. Consider the relation represented by this table:

1. Describe the relation in words.

The relation shows the association

“_____________________” from a set of

__________________ to a set of

_________________________.

2. Represent this relation:

a. As a set of ordered pairs

b. As an arrow diagram.

What happens if the elements of one or both sets are numerical? All the interpretations are the same, however, you are now in a position to graph the relationships as well.

Ex. Different breeds of dogs can be associated with their mean heights. Consider the relation represented by this graph.

a. Represent this relation as a table.

b. Represent this relation as an arrow diagram.

Homework from the Calendar

Section 5.2 Functions Math 10C

New words!!!!

The DOMAIN is the set of first elements (the x-values) The RANGE is the set of the second elements (the y-values) A FUNCTION is a special type of relation where each element in the domain has

ONLY ONE value in the range. (you never see a repeat in the first set or x)

Which of the following is a function?

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.

Input Output

1

9

6

10

What can a table show you?

In the workplace, a person’s gross pay, P dollars, often depends on the number of hours worked, h. The pay DEPENDS on the HOURS.

You Try:

The table to the right shows the masses, m grams, of different numbers of identical marbles, n.

a) Is this relation a function? Why or why not?

b) Identify the independent and dependent variables.

c) What is the domain and range?

Putting it all together in a more mathematical format means that we put the relation into an equation. It is designed in a specific method called FUNCTION NOTATION.

Ex. The mass of 1 quarter is 4.4grams.

- This means that the input is q quarters, the output is M mass in grams.- Since M is a function of q, we can write this equation as:

M(q) = 4.4q

This format allows us to determine the mass of any number of quarters.

M(10) = 4.4q M(25) = 4.4q M(71) = 4.4q

Any function that consists of 2 variables can be written using function notation. For example:

In science d = 4t + 5

In Math d(t) = 4t + 5

When we write an equation that is not related to a context as above, we use x as the independent variable and y as the dependent variable.

Function Notation in this case looks like this….

You Try:

The equation C = 25n + 100 represents the cost (C in dollars), for a feast following and Arctic Sports competition, when n is the number of people attending.

a) Write the equation in function notation.

b) Determine the value of C(100). What does the value represent in the context of the question?

c) Determine the value of n when C(n) = 5000. What does the value represent in the context of the question?

d) Determine the value of n when C(n) = 5200

Homework from the Calendar

Section 5.3 Interpreting Graphs Math 10C

TODAYS LESSON IS AN INDEPENDENT WORKSHOP LESSON

What this means is, I will provide you with all necessary information, but I will not be instructing to the lesson. It is up to you to draw the necessary conclusions and information as best as you can. I will clarify any and all problem areas tomorrow when we take up the homework….so be sure to ask questions!!!!!

Investigation – with a partner

This graph shows the depth of water in a bathtub as a function of time.

a. What does each segment of the graph represent?

0 – A

A – B

B – C

C – D

D – E

E – F

F – G

- Compare your description with another group. Are the stories the same or different? What should they be? Why?

b. Sketch and label a graph to represent the following situation.You put a plug in the bathtub and turn on the taps.You leave the bathroom and return to discover that the bath has overflowedYou turn off the taps and pull out the plug to let some of the water out.You put the plug back in

- Compare your graph with another group’s. How are they the same? How are they different?

The properties of a graph can provide quite a bit of information

Putting it all together.

Each point on the graph represents a bag of popcorn. Ful ly answer the questions.

a. Why are the dots not connected?

b. Which bag is the most expensive and what does it cost?

c. Which bag has the least amount of mass and what is that mass?

d. Which bags have the same mass and what is that mass?

e. Which bags have the same price and what is that price?

f. Which of bag C or bag D has the better value for its money? Defend your decision mathematically.

Do not forget that the graph provides you with a story as to what is happening. It is your job to be able to interpret that story.

Describe the journey for each segment of the graph to the right. Be as detailed as the information allows.

You are also going to have to take an anecdotal and graph it. See if you’re able to do that with this..

Samuel went on a bicycle ride. He accelerated until he reached a speed of 20km/h, then he cycled for 30 minutes at approximately 20km/h. He arrived at the bottom of a hill, and his speed decreased to approximately 5km/h for 10 minutes as he cycled up the hill. He stopped at the top of the hill for 5 minutes to catch his breath.

1. What is the dependent variable? _____________________

2. What is the independent variable? ____________________

3. Sketch and label a graph depicting speed as a function of time.

Homework from the CalendarMath Lab Assignment tomorrow Due at end of Class

Section 5.5 Graphs of Relations and Functions Math 10C

There is no relevant reason for this being here….I just thought it was funny!!!!

What we already know….

1. A RELATION is a rule that relates elements of one data set to the elements of another data set. ALL GRAPHS are relations.

2. A FUNCTION is a special type of relation where there is EXACTLY one element of the second data set. Not all graphs are functions.

We can see if a graph is a function by looking for a few things:

1. There are no two points on the same vertical line of the graph.2. Use the “Vertical Line” test with your pencil…..

When we graph…..

Determine the Domain and Range for each…

Up to this point, each graph or relation has only had individual points. What happens to the Domain and Range if the points are connected?

How do you know if we should draw a discrete or continuous graph?

Typically there are 3 different scenarios regarding Domain and Range…

1. Relation is continuous.

2. Relation is bound.

3. Relation is discrete

Try these:

Homework from Calendar

Section 5.6 Linear Relations Math 10C

A LINEAR RELATION is when a graph of a straight line is formed. This means the rate of change is CONSTANT.

What is RATE OF CHANGE?

In science, you would have learned about rate of change when graphing distance, speed and time graphs. Did the car speed up at the same rate? Is the car traveling at the same speed? To determine this on a graph we look at the difference between two points.

Rate of Change = Change in Y (Rise) Change in X (Run)

What if you are given a table of values instead? How would you determine the rate of change?

Determine whether or not the following tables represent linear or non-linear functions.

If the cost of a rental car is $60 plus $20 for each 100km driven then:

a. The independent variable is _____________________.

b. The dependent variable is _______________________.c. Create a table of values to represent the cost of driving a car from 0 – 500km.

d. Graph the relation.e. Is this relation a function?

f. Determine the rate of change.

You Try:

Homework Pg. 308 #3 – 5, 7, 12**

Section 5.7 Interpreting Linear Relations Math 10C

Graph the function complete a table of values and then plot the points.

How about using that fancy calculator of yours????

Using your calculator determine the following:

Homework from Calendar