1. Math 374
Graphs

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

3. 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.

4. Notes
We recall

5. 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

6. 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

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

8. 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

9. 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…

10. . . . . Q2 (-,+) Q1 (+,+) Q4 (+,-) Q3 (-,-) Plotting B (-4, 2)
A (3,1)
B (-4,2)
C (-4, -4)
D (2, -2) . .
B (-4, 2)
(A 3, 1) .
Q4 (+,-)
Q3 (-,-) .
D (2, -2) .
C (-4, -4)

11. Standard Form Method Slope Y intercept
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

12. Identifying Slant and Slope
Y Int
Slant
Slant
Y Int

13. Standard Form Dependent Variable (DV) Y Intercept or Starting Value
Recall y = mx + b
Dependent Variable (DV)
Y Intercept or Starting Value
Slope
Independent Variable (IV)

14. 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

15. Rise, Run & Slope
Slope
Slope
Rise
Rise
Run
Run

16. 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

17. 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)
Rise
Run

18. 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…

19. Ex#1: y=2x+3 Question: Draw this line (1,5) 0,3
Where can you plot the y intercept?
0,3
What is the y intercept?
What is the slope
What does the slope mean?
Up 2, Right 1

20. Example #2 y = -5 x
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)

21. 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

22. 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

23. Graphing with Scale
Scale is mostly used to make sure your graph can be seen
Consider y = 2x + 100 3

24. Ex#5 y=2x+100 3 How will you measure m = 2/3? Y intercept? (300,300)
500 x y
How will you measure m = 2/3?
Y intercept?
(300,300)
Slope?
(0,100)
You can put 500 along the x axis which means each hash mark is 100
Note slope is a ratio so scale does not effect it

25. Ex. #5 200x + 300y =
300y = - 200x
Y = -2x + 400 3
Plot it
Do #4 on stencil use form C

26. 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)

27. Consider y = 2x – 5 3 Stencil: Do #5
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

28. Inequality Graphs
The straight line of the graph divides the plane into two regions
One side will be greater than, one side less than

29. 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

30. Ex y > x + 3 Step 1: Draw Line Y intercept? Slope?5 x y
Y intercept?
Slope?
m = 1 (up 1, right 1)
Dotted Line or solid?
Shade above or below?

31. Ex y < x + 3 Step 1: Draw Line Y intercept? Slope?5 x y
Step 1: Draw Line
Y intercept?
Slope?
m = 1 (up 1, right 1)
Dotted Line or solid?
Shade above or below?

32. Ex 5x - 10y < 30 Step 1: Put in Standard Form 10y < - 5x + 302
y intercept?
Slope?
m = 1 (up 1, right 2)
Dotted Line or solidline?
Do #6 in C
Shade above or below?

33. Point of Intersection If we have two graphs, we create four regions 12 4 3

34. Consider y > 3x – 5 y < -2x + 5 Draw lines… one at a time
2nd line…
y intercept of 1st?
y int?
Slope?
Slope?
Dotted / solid?
m = 3 (3 up, right 1)
Above or Below?
Dotted Line or solid?
Use arrows
Shade above or below?
Hint… with 2 lines, use arrows at first instead of shading
Shade where they intersect!

35. 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