RELATIONS LEVEL 3

1. RELATIONS LEVEL 3 UNDERSTANDING GRAPHS LINEAR RELATIONS

2. In these notes • Cartesian plane • Variables in a Relation • Table of values • Some types of relations that you must know • Working from a word problem to a graph • Working from a graph to an equation

3. The Cartesian Plane (graphing) • You need to know the parts of a graph. • You need to label these parts to get all marks.

4. Can you write and draw all of these? Title of graph

5. Selecting a scale for x-axis and y-axis • Try to fit the scale that best fits the graph. In this case, the scale of x is 1 and the scale of y is 5.

6. Variables in a relation • A line of a graph is made of a series of points. • Each point is made of an x-value and a y-value • The x and y are related mathematically. • The x variable is called the independent variable. • The y value is called the dependent variable. • For each value of x on a graph there is a value of y. • You will be asked to read a problem and identify the dependent and independent variable.

7. Example #:1 Variables in a Relation • Last January 1st, Myriam’s bank account had a balance of $500. Since then, she has deposited $40 a week into her account. The situation is represented by the following equation. y = 40x + 500 What is the independent variable? Is it the $500? Is it the money in the bank account? Is the $40? Is it the number of weeks?

8. Example #1 Variables and relations y = 40x + 500 y = total of money and y is the dependent variable x = number of weeks and x is the independent variable

9. What to know: Variables and Relations • The dependent variable (y) is usually the answer being asked for. It depends on the value of the x variable. • The dependent variable in a word problem is called the total, the cost, the outcome, the result, the consequence or another similar term. • We use x and y variables in a table of values to help us understand what the relation is.

10. Tables of Values A table of values can be vertical or horizontal. Here are examples of the same table in the two formats. Can you figure out the relation?

11. Tables of Values A table of values is used to plot a graph and then to draw a line through the points. Below is an example for y = 5x + 4

12. There are several types of relations • For this unit, you must memorize the following: • Direct • Indirect or partial • Inverse • Zero • To the square

13. Types of Variations • Direct: Straight line y = mx +b where b = 0

14. Types of Variations • Partial: Straight line y = mx +b where b 0

15. Types of Variations • Inverse: A relationship between two variables in which the product is aconstant. When one variable increases the other decreases in proportion so that the product is unchanged. (In this example the product of x and y is always 60.)

16. Types of Variations • Zero: The value of y is a constant. The value of x changes but the value of y does not. This means that m = 0

17. Types of Variations • Square: The value of y is the a multiple of the square of x. In this situation y = 5 * x2

18. Going from problem to graph • Read the following problem: The following were the rates charged by an Internet provider in 2006: Basic monthly rate is: $15 up to 12 hours of use Rate for additional hours is: $1 for 1 hour Determine the rule for this relation.

19. Going from problem to graph • What are the dependent and independent variables? Total money = dependent Number of hours = independent Constants are the rate of change (m) and the basic rate (b).

20. Going from problem to graph • Create a table of values • What is occurring? • There are two types of relations. Can you spot them?

21. Going from problem to graph First part of the graph: When 0 x 12 y = 15 a = $0 x = number of hours b = $15 What type of relation is this? Second part of the graph: When 12 < x y = $1x + 15 a = $1 x = number of hours b = $15 What type of relation is this?

22. Going from problem to graph Label the different parts of the graph with the values of the tables of values. If you are not sure, ask your teacher.

23. Going from problem to graph Label the different parts of the graph with the values of the tables of values. y-axis is cost Goes up by one for each hour after 12. $15 x-axis is hours Up to 12 hours It costs the same.

24. Going from graph to solution • What is the equation that represents the line?

25. Going from graph to solution • First, it is a straight line? Does it go through the origin? • What type of relation is it?

26. Going from graph to solution • Since it is a partial variation, we know that b is equal to the y-intercept. The value of y when x is zero. This means b = 5

27. Going from graph to solution • Find m (rate of change or the slope) in y = mx + b Select any two points on the line (0,5) and (2,15)

28. Going from graph to solution • We calculated b and m so the equation for this line is y = 5x + 5 Calculate y by using x = 4 and then x = -2. Do you get the correct answers for y? Check by looking at the graph.

29. Practice • Know how to read and plot a graph • Know the different types of relations • Know how to calculate a and b in y = ax + b • Know that y = ax + b or y = b + ax are the same. • Know how to create a table of values and find the missing values.

30. You have just studied • Cartesian plane • Variables in a Relation • Table of values • Some types of relations that you must know • Working from a word problem to a graph • Working from a graph to an equation

31. The End • Study: Know the terms, know the graphing skills • Practice the problems done in class. • Ask questions when you do not get the correct answers.