Algebra 1 > Notes > Yorkcounty Final > Yc > Lesson 18

Lesson 18 Relations and Functions NOTE: Graph Paper Is Needed for This Lesson

Graphing Ordered Pairs (A Review) In pre-algebra you learned to graph ordered pairs in a coordinate plane with an x-axis and y-axis. An ordered pair contains two coordinates (x, y). The x number tells you where to go on the x-axis and the y-number tells you where to go on the y-axis. The x-axis is positive to the right and negative to the left. The y-axis is positive to the top and negative to the bottom. They are like two intersecting number lines.

For example, if you were asked to graph the coordinates ( 3 , -2) you would go to positive 3 on the xaxis and to negative 2 on the yaxis. Where these two lines meet would be the coordinate point of your dot…. Another way to think of coordinate points is that the xcoordinate tells you how far to move left or right and the y-number tells you how far to then move up or down.

The quadrants of the coordinate plane are listed. Quadrant 1 (I), 2 (II), 3(III) and 4 (IV).

y II

I

x III

IV

Definition:

Relation

A relation is any set of ordered pairs. The domain of a relation is the set of all first coordinates (xcoordinates) of the relation.

The range of a relation is the set of all second coordinates (ycoordinates) of the relation. (2, 3) (4, 5) (1, 9) Domain : 2, 4, 1

Range: 3, 5, 9

Domain refers to x values (also called input) Range refers to y values

(also called output)

Graph the following relation : { (1,1) (2,3) (0, -2) (3,-1) (-5, -3) (-5, 5)} y What is the domain and the range?? (you don’t have to repeat values, ex for the domain you do not have to repeat –5, only list it once)

Domain (x values) {-5, 0, 1, 2, 3} Range (y values) {-3, -2, -1, 1, 3, 5}

Try It:

Graph the relation on a piece of graph paper. Then state the Domain and the Range.

{ (-1, 2) (-4, 3) (2,2) (4, 1) (0, -1) (3, -2) }

Solution Domain : ( -4, -1, 0, 2, 3, 4) Range: (-2, -1, 1, 2, 3)

All of the points are in either quadrant I, II, or IV. No points lie in quadrant III.

Also note that a relation can be given as a table. R = { (5, -5) (-2, 0) (4, 0) (-1, -3) (-4, 3) } Here are a few ways of expressing this relation. X

Y

X

5

-2

4 -1

-4

5

-5

Y -5

0

0 -3

3

-2

0

4

0

Y -5

0

0 -3

3

-1

-3

X

-2

4 -1

-4

-4`

3

5

Relations can also be expressed by an equation. We can use a graphing calculator to help us graph the equation and to even generate a table. You will need to get your calculator now. Instructions will be given for a TI-83.

Let’s start with the equation y = 3x -2. Press the y= button on your calculator. It is a blue button in the top left corner. You should get a menu that says Plot 1, Plot 2, Plot 3, y1, y2, y3…etc…

Make sure the Plots (Plot 1, Plot 2 and Plot 3) are not highlighted. If they are, arrow up to them, press enter and then arrow away.

Type in 3x - 2 at Y1. To get the x, you press the button on your calculator that says x,θ,T . You now have this relation stored in your calculator.

Now that we have the equation stored, we can look at a graph of the equation. There are two ways to do this. You can press the graph button in the top row or press zoom and 6 .

When you press zoom and 6 this makes what is called a standard graph. Where the x- and y-axis automatically go from -10 to 10 each. If you want to see other values on your graph, you can press the window button and adjust the maximum and minimum values of your graph. You can also adjust the scale on your x- and y-axis. However, never change x-res. Always leave it at 1. Take a few moments to explore these options.

When you are done exploring--graph the line on zoom 6, and advance to the next slide to check your line.

y

x

Your graph may not look exactly like this one because you have both axes going -10 to positive 10 and the tickmarks on the y-axis are closer together than on the x-axis. If you want your graph to look like it was graphed on a square grid-try pressing zoom and choosing the option that says “square”

What if we wanted to see a table of x-and y-values for this relation?

You can do this by pressing 2nd and Graph because it says table above the graph button. You should get a table of values. Use your up and down arrows to check for the y-values at the following x-coordinates: y = 3x – 2 X

Y

-2

-8

10

28

17

49

Remember Domain also means input and Range also means output. Think of it this way. If you take x= -2 and INPUT into the function you get an OUTPUT of –8.

FUNCTIONS

Now that we know a relation is any set of ordered pairs, we need to define function. A function is a relation in which no two ordered pairs have the same first coordinate. In other words, no domain value (that is no x-value) can be repeated.

{ (1,6) (8, -3) (-1,-6) (8,2) } Is this a function?? Look at the ordered pairs and see if any of the x-values is repeated. If the x-value is repeated then it is NOT a function. Y-values can be repeated. NO. This is not a function because 8 is repeated twice.

Which is a function? A)

X 1 4 6 6

B)

Y 3 2 5 3

C)

X 3 5 2 8 Y 1 6 2 1

X 2 5 3 5 Y 4 1 5 2

D)

X 0 4 0 2 Y 8 2 4 3

Choice C is the only function. Both A, B and D have repeated values for X.

Practice 1) Is the relation { (-1,5) (3, -8) (2,-5) (9,-8) } a function? Why? 2) Which of the tables below represents a function. Explain Why?

A)

X

1 0 0 7

Y

4 1 4 9

B)

Check your work on the next slide

X

3 7 4 9

Y

8 2 5 5

1) Yes. No X values are repeated. 2) A is a function. B is not a function because the X value of 3 is repeated twice.

Function Notation: y = 5x is a function. We sometimes use a special notation instead of using the variable y. This is called function notation: f(x). f(x) = 5x

y = 5x

Using f(x) = 5x lets us know that 5x is a function. y = 5x is a function but we do not know that by looking at the equation. We would have to find out that it is a function. Also, we can find the value of a function at a certain x value by asking for f(x) and putting a number instead of x in the parentheses--as you will see in the next example.

f(x) = x2 - x. Find f(-3) NOTE: Remember when you substitute make sure you use parentheses Solution: f(-3) means to substitute -3 for x in the equation x2 - x. There are two ways we can get the answer: Way 1: Substitute with pencil and paper method: f(-3)

in

f(x) = x2 - x f(-3) = (-3)2 - (-3) you can evaluate in calculator = 9 +3 = 12

Way 2: Substitute by using a calculator Put in x2 - x in the y= menu on your calculator. Then press 2nd graph and look at a table of values. Find the y-value when the x-value is -3 by scrolling up and down the table. It should be 12.

Try It

f(x) = 2x2 + 5x

Find f(2)

Solution f(x) = 2x2 + 5x Find f(2) f(2) = 2(2)2 + 5(2) = 2(4) + 10 = 8 + 10 = 18 18 is the solution

What is the range of the function f(x) = -x2 - 3 when the domain is {-4, -1, 5}?

Solution: by using the TI-83 Graphing calculator we can save time. Press y = and type in the equation at y1= -x2 - 3x Press 2nd and

Graph

to see a table of values

Scroll up and down the table and look at the y-values for the x-values of -4, -1 and 5. They are -19, -4, and -28

Review – Relations and Functions A relation is any set of ordered pairs. A function is a relation in which no two ordered pairs have the same first coordinate. f(x) is called function notation. This allows us to know that the operation is a function. f(x) = 10x2 ; thus 10x2 is a function. Domain refers to the x-variables in a relation/function Range refers to the y-variable in a relation/function

JUNK y