Math 374

Graphs

Topics

Cartesian Plane Methods of Graphing Intercept Slope Scale First Quadrant Inequality Graphs Region

Cartesian Plane

Named after Rene Deo Cartes a french mathematician

Also a philosopher “I think therefore I am” His goal was to create a “picture” that could

show a relationship between two variables. We have one for one variable – the number line.

Notes

We recall

- 2 -1 0 1 2 3

Some Facts

We only need two points to draw a straight line

The point where a graph crosses or touches the x axis is called the x intercept

It is found by substituting y = 0 The point where a graph crosses or touches

the y axis is called the y intercept It is found by substituting x = 0

Intercept Method

Calculate both intercepts. Place on graph and join

Example #1: y = 2x – 6 X intercept (y = 0) 0 = 2x – 6 -2x = - 6 x = 3

Intercept Method

Now find Y intercept Example #1: y = 2x – 6 Y intercept (x = 0) y = 2 (0) – 6 y = -6

Finding X and Y Intercept

Example #2: 5x – 3y = 15 x int (y = 0) 5x – 3(0) = 15 5x = 15 x = 3 y Int (x = 0) 5(0) – 3y = 15 -3y = 15 y = -5

Drawing on Graph

Now that you know the x & y intercept, you have two points and now can draw the straight line… do it!

Practice plotting with other points…

1 2 3 4 5-1-2-3-4-5

1

2

3

4

5

-1

-2

-3

-4

-5

x

yPlottingPlotting

. C (-4, -4)

.

.

.. B (-4, 2)

D (2, -2)

(A 3, 1)

Q1 (+,+)

Q4 (+,-)Q3 (-,-)

Q2 (-,+)A (3,1)

B (-4,2)

C (-4, -4)

D (2, -2)

Standard Form Method

All straight lines have a y intercept and a slant called a slope.

If the relationship is in standard form we can write it…

y = m x + b

Slope Y intercept

Y Int

Slant

Slant

Y Int

Identifying Slant and Slope

Standard Form

Recall y = mx + b

Dependent Variable (DV)

SlopeIndependent Variable (IV)

Y Intercept or Starting Value

Relationship of y & b

It is easy to see how b is the y intercept; we substitute x = 0

x = 0 y = m(0) + b y = b

Rise, Run & Slope

Slope

Rise

Run

Rise

Run

Slope

Understanding the Slope

If m or the slope is 2 this means a rise of

2 and a run of 1 (2 can be written as 2 )

1 If m = - 5, this means a rise of -5 and right 1 If m= -2 this means rise of -2 right 3

3

Understanding the Slope

Consider m = -3 4 What is the rise and what is the run? Suggest to put the negative sign on the top

to clarify (rise of -3) Numerator always rise (could go up or down) Denominator always run (right only)

RiseRun

Consider y = 2x + 3

What is the slope, rise, run and y intercept? We have a slope 2 2 can be written as 2

1

Rise of 2

Run of 1

y intercept of 3 (y = b)

Plot on graph paper the following…

1 2 3 4 5-1-2-3-4-5

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

Ex#1: y=2x+3Ex#1: y=2x+3

0,3

(1,5)

Question: Draw this line

What is the y intercept?

What is the slope

What does the slope mean?

Where can you plot the y intercept?

Up 2, Right 1

Example #2 y = -5 x + 1 7

What is the y intercept, slope? Rise and run? Y intercept is 1 Slope is -5/7 Rise is – 5 Run is 7 Plot on graph (put it on graph paper)

Example #3 y = x

What is the y intercept, slope, rise and run? y intercept = 0 (y int let x = 0) Slope = 1 Rise of 1 Run of 1 Plot on graph

Example #4: 3x – 4y = 12

What is the y intercept, slope, rise and run? Must put in standard form -4y = - 3x + 12 y = 3x – 3 4 y intercept = -3 Slope ¾ Rise of 3, run of 4 Plot on graph

Graphing with Scale

Scale is mostly used to make sure your graph can be seen

Consider y = 2x + 100

3

Ex#5 y=2x+100 3Ex#5 y=2x+100 3

(0,100)

(300,300)

500

x

y

You can put 500 along the x axis which means each hash mark is 100

Y intercept?

Slope?

How will you measure m = 2/3?

Note slope is a ratio so scale does not effect it

Ex. #5 200x + 300y = 120000

300y = - 200x + 120000 Y = -2x + 400

3 Plot it Do #4 on stencil use form C

1st Quadrant

There will be times when you will need to put the graph only in the 1st quadrant

The problem only exists when the y intercept is negative

In that case, work with the x intercept (sub y = 0)

Consider y = 2x – 5 3

Show how the graph intersects in the 1st quadrant Notice that b is negative. In those cases, work with x int (let y = 0) 0 = 2x – 5

3 0 = 2x – 15 -2x = -15 x = 7.5

Stencil: Do #5

Inequality Graphs

The straight line of the graph divides the plane into two regions

One side will be greater than, one side less than

The Trick in Standard Form

If greater then shade above > If less then shade below < If equal then solid line If not equal then dotted line

Ex y > x + 3 Ex y > x + 3

5

x

y

Y intercept?

Slope?

Step 1: Draw Line

m = 1 (up 1, right 1)

Dotted Line or solid?

Shade above or below?

Ex y < x + 3 Ex y < x + 3

5

x

y

Y intercept?

Slope?

Step 1: Draw Line

m = 1 (up 1, right 1)

Dotted Line or solid?

Shade above or below?

Ex 5x - 10y < 30 Ex 5x - 10y < 30

5

x

y

y intercept?

Slope?

Step 1: Put in Standard Form

m = 1 (up 1, right 2)Dotted Line or solidline?

Shade above or below?

-10y < - 5x + 30

y > 1x – 3

2

Do #6 in C

Point of Intersection

• If we have two graphs, we create four regions

1

3

4 2

Consider y > 3x – 5 y < -2x + 5 Consider y > 3x – 5 y < -2x + 5

5

x

y

Draw lines… one at a time

Slope?

Hint… with 2 lines, use arrows at first instead of shading

m = 3 (3 up, right 1)

Dotted Line or solid?

Shade above or below?

y intercept of 1st? 2nd line…

y int? Slope?

Dotted / solid?

Use arrows

Above or Below?

Shade where they intersect!

Find POI (Point of Intersection), you can also use equations

y = 3x – 5 y = -2x + 5 3x – 5 = -2x + 5 5x = 10 x = 2 x = 2 y = 3(x) – 5 y = 3 (2) – 5 y = 1 POI (2, 1)

Do 7 in E

Finish Study Guide