GRAPHISH: THE LANGUAGE OF GRAPHS

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GRAPHISH THE LANGUAGE OF GRAPHS
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(No Transcript)
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Graphing A brief review

• Variables Xs and Ys
•  
• Functions Yf(X)
•  
• Y is a function of X
•  
• Y dependent (endogenous) variable
•  
• X independent (exogenous) variable.

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Number Lines
40
30
20
B
10
A
0

- 3
- 2
- 10
- 20
Horizontal number line
- 30
- 40
Vertical number line
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Coordinate Geometry
Y
4 3 2 1
(X, Y)
X
-4 -3 -2 -1 1 2 3 4
O
-1 -2
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Coordinate Geometry

• The figure above shows the (rectangular)
  coordinate plane. The horizontal line is called
  the x-axis and the perpendicular vertical line is
  called the y-axis. The point at which these two
  axes intersect, designated O, is called the
  origin. The axes divide the plane into four
  quadrants. 1,2 ,3 and 4, as shown.

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Coordinate Geometry

• Each point in the plane has an x-coordinate and a
  y-coordinate. A point is identified by an ordered
  pair (x, y) of numbers in which the x-coordinate
  is the first number and the y-coordinate is the
  second number.
• (4,5) means that the point is 4 units to the
  right of the y-axis ( that is x4) and 5 units
  above the x-axis ( that is y5). The origin has
  coordinates (0,0)

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Coordinate System
40
30
B
(1, 20)
20
A
10
(4, 5)
0
1
2 3
4
5
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From Table to Graph
Quantity of pens
3.00
2.50
2.00
Price of pens (in dollars)
1.50
1.00
.50
0
1 2 3 4 5 6 7 8
Quantity of pens bought
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Linear and Nonlinear Curves
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5

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Price (in dollars)
Price (in dollars)
3
2
1
0
10
20
30
40
Quantity
Quantity
Linear Curve
Nonlinear Curve
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Graphs

• (X,Y)
• Independent variable is measured on the
  horizontal axis
• Dependent variable is measured on the vertical
  axis.
• Slope (straight-Line Graphs)
• tells how much Y will change every time X
  changes by one unit
• slope of a straight line
• change in vertical variable/change in
  horizontal variable.
• rise / run
• Linear Equations
• Y a bX
• a is vertical intercept, it determines the
  graph position.
• b is the slope. It can be positive, negative
  or 0.

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Equations and Graphs
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Y

• A linear graph can be expressed as an equation in
  the form y mx b where m is the slope and b
  is the vertical intercept.
• The slope of the blue line is 8/-4 - 2 and the
  intercept is 8, so the equation is y - 2x 8.
• The slope of the red line is 8/-8. The intercept
  is 8, so its equation is y -x 8.

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8
7
6
5
4
3
2
1
0
2
1
4
6
3
5
7
8
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Slopes of Curves
c
10
Slope 1
9
d
Rise 1
A
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Slope 4
Run 1
Slope - 4
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Rise 4
6
Rise - 4
5
B
4
Run 1
Run 1
e
Slope -0.5
3
L
a
Rise -1
2
Slope 1
Run 2
E
b

1
Rise 1
e
Run 1
0
1 2 3 4 5 6
7 8 9 10 11
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Positive relationship v.s. Negative relationship

• Suppose YGPA
• Xnumber of hours spent
  studying per week
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• As X increases, what do you expect to happen to
  Y?
• As X increases, Y increases.
• positive (or direct) relationship
• Suppose YGPA
• Xnumber of hours spent
  watching TV
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• As X increases, what do you expect to happen to
  Y?
• As X increases, Y decreases.
• negative (or indirect) relationship

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Inverse and Direct Relationships
Direct relationship When X goes up, Y goes
up When X goes down, Y goes down
Inverse relationship When X goes up, Y goes
down When X goes down, Y goes up
X
X
Y
Y
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Example telephone bills

• Let Ycost of an international phone call
• Let Xlength of call in minutes 
• 2 initial connection fee
• 50 cents per minute
• functional relationship between X and Y
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• Y f(x) 2 .5 X

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• Pick up the values of X and Y

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Solve for the linear equation

• Step (1) Pick any two points.
• (X2,Y2) (1, 2.5) (X1,Y1) (0, 2)
• Step (2)  Plug into formula
• slope ?Y/?X
• (Y2 - Y1)/(X2 - X1)
• (2.5 2)/(1 0)
• .5/1
• .5
• INTERPRETING THE SLOPE
• Slope tell us what happens to Y as X increases by
  ONE unit.
• A Linear Equation 
• X-intercept
• Y-intercept
• Equation of a line Y a bX
• In our PHONE CALL EXAMPLE
• a 2 and b .5
• So, Y 2 .5 X

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Straight Lines with Different Slopes and Vertical
Intercepts
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Straight Lines with Different Slopes and Vertical
Intercepts
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Example on advertising and sales
Table A.1 Advertising and Sales at Len Harrys
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Example on advertising and sales
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Solve for the straight line function

• First, calculating slope
• Step 1, pick up 2 points
• (X1,Y1)(2,46), (X2,Y2)( 6,58)
• Step 2, plug into formula
• Slope (Y2 - Y1)/ (X2 X1)
• (58-46)/(6-2)
• 12 / 4
• 3
• b
• So Y a 3X, plug in any point value for X
  and Y. e.g. (2,46)
• 46 a 3x2,
• So a 46 6 40,
• And Y 40 3X

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Presenting Information Visually
(a) Line graph
(c) Pie chart
(b) Bar graph