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

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 $100 200 300 400 Income (Y) LO7 1-2

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 LO7 1-3

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 LO7 1-4

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 $100 200 300 400 Income (Y) LO7 1-5

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. LO7 1-6

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 4 8 12 16 20 Attendance LO7 1-7

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. LO7 1-8

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 LO7 1-9

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. LO7 1-10

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 +.5. 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). $100 200 300 400 Income (Y) LO7 1-11

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 -2.5. 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. 4 8 12 16 20 Attendance LO7 1-12

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. LO7 1-13

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. 5 10 15 20 LO7 1-14