1. 01A Appendix Limits, Alternatives, and Choices
In this appendix, we will cover the fundamentals of graphing, including how to create a graph and the relationships between variables. We will discuss how direct and inverse relationships look graphically, and by the end of the appendix, you should be able to look at a graph and immediately identify whether it demonstrates a direct or inverse relationship. The equation of a line will be discussed, along with how to find the slope of linear and nonlinear graphs.
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
McGraw-Hill/Irwin

2. Construction of a Graph
Table
Income
Consumption
Point
$ 0
100
200
300
400
$ 50
100
150
200
250 a b c d e
$400
300
200
100
Consumption (C) e
Each pair of values in the table represents one point on the graph. d c b a $
Income (Y)
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3. Direct and Inverse Relationships
Direct relationship
When looking at the graph, a direct relationship has an upward sloping line. This means that when looking at the graph from left to right, the line moves up, or away from the horizontal axis. You should be able to look at a graph and immediately recognize if it is a direct relationship.
Both variables move up or down together
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4. Direct and Inverse Relationships
An inverse relationship graphs as a downward sloping line. This means that when looking at the graph from left to right, the line moves down or gets closer to the horizontal axis.
Variables move opposite of each other
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5. Positive slope of a straight line
Slope of a Line
Positive slope of a straight line
Vertical Change
+50 1 2
Slope = = =
= 0.5
Horizontal Change
+100
$400
300
200
100
Consumption (C)
When calculating the slope of a line, you can use any two points on the line to calculate the change because the line has a constant slope. You need to be careful and consistent when calculating the vertical and horizontal changes to be sure that you get the correct sign (positive or negative). A direct relationship always has a positive slope. 50
Vertical
Change
100
Horizontal Change $
Income (Y)
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6. Slope of a Line $50 40 4 30 8 20 12 10 16 Ticket Price
Attendance, thousands
$50 40 4 30 8 20 12 10 16
Using the data on ticket price and attendance, we can graph the following line.
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7. Negative slope of a straight line
Slope of a Line
Negative slope of a straight line
Vertical Change
-10 1 2
Slope = = -2
-2.5 = = 50
4030 20 10
Horizontal Change +4
Ticket Price $
Vertical
Change
-10
An inverse relationship always has a negative slope because the variables move in opposite directions. 4
Horizontal Change
Attendance
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8. Slopes and measurement units Slopes and marginal analysis
Slope of a Line
Slopes and measurement units
Slopes and marginal analysis
Infinite and zero slopes
The slope of a line can vary based on the units that are used to measure the slope.
Slope is a way of illustrating marginal analysis. Because the denominator is the change in the horizontal axis, the slope measures how much more or less is gained with the variable on the vertical axis when there is a 1 unit change on the horizontal axis.
When two variables are not related, they will have an infinite or zero slope.
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9. Slope of a Line Slope = Zero Slope = Infinite Price of Bananas
Consumption
The graphs above show that the variables are unrelated. No matter what the price of bananas might be, the number of watches purchased remains the same. This is because the price of bananas does not affect the number of watches purchased.
There is no relationship between consumption and the divorce rate. Consumption remains the same no matter how high or low the divorce rate is.
Purchases of watches
Divorce Rate
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10. Equation of a Linear Relationship
y = a + bx
y is the dependent variable
a is the vertical intercept
b is the slope of the line
x is the independent variable
This is the equation of a line and the pieces that make up the equation.
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11. Equation of a Line Y = 50 + .5C $400 300 Consumption (C) 200 100
Consumption (C)
We calculated the slope of this line earlier, and we found that the slope is We can see from the graph that the vertical intercept is 50. Therefore the equation of this line is:
Y (income) = 50 (vertical intercept) + .5C (where .5 is the slope and C is the consumption). $
Income (Y)
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12. Equation of a Line
P = 50 – 2.5Q 50
4030 20 10
Ticket Price
We calculated the slope of this line earlier, and we found that the slope is We can see from the graph that the vertical intercept is 50. Therefore the equation of this line is:
P (price) = 50 (vertical intercept) - 2.5Q (where -2.5 is the slope and Q is the quantity). In this case, the slope is being subtracted because the slope is negative.
Attendance
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13. Slope of a Nonlinear Curve
Slope always changes
Must use a line tangent to the curve to find slope at that point
When calculating the slope of a nonlinear curve, the slope is always changing. This means the slope will be different at every point on the curve. In this case to find the slope, you must draw a line tangent to the curve and calculate the slope of the line by finding two points along the tangent line.
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14. Slope of a Nonlinear Curve
Use tangent lines to find the slope 20 15 10 5
The slopes of these two points on the curve will be different from each other.
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