Graphs’ purposes: ◦ Visually express ideas that might be less clear if described with equations or words Q(p) = a + b*p ◦ Powerful way of finding and interpreting patterns Increasing / decreasing over time? 1 Graphs of two variables: the coordinate system ◦ Display two variables on a single graph ◦ Scatterplot ◦ Ordered pairs of points x-coordinate Horizontal location y-coordinate Vertical location © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use. 2 http://www.nytimes.com/interactive/2013/08/21/us/household-incomes-slowclimb.html?ref=politics CHAPTER 1 APPENDIX How to Read and Understand Graphs A graph is a two-dimensional representation of a set of numbers, or data. Time Series Graphs A time series graph shows how a single measure or variable changes over time. TABLE 1A.1 Total Disposable Personal Income in the United States, 1975–2012 (in billions of dollars) Year 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 Total Disposable Personal Income 1,187.3 1,302.3 1,435.0 1,607.3 1,790.8 2,002.7 2,237.1 2,412.7 2,599.8 2,891.5 3,079.3 3,258.8 3,435.3 3,726.3 3,991.4 4,254.0 4,444.9 4,736.7 4,921.6 Year 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Total Disposable Personal Income 5,184.3 5,457.0 5,759.6 6,074.6 6,498.9 6,803.3 7,327.2 7,648.5 8,009.7 8,377.8 8,889.4 9,277.3 9,915.7 10,423.6 11,024.5 10,772.4 11,127.1 11,549.3 11,930.6 FIGURE 1A.1 Total Disposable Personal Income in the United States: 1975–2012 (in billions of dollars) Graphing Two Variables X-axis The horizontal line against which a variable is plotted. Y-axis The vertical line against which a variable is plotted. origin The point at which the horizontal and vertical axes intersect. Y-intercept The point at which a graph intersects the Y-axis. X-intercept The point at which a graph intersects the X-axis. Using the Coordinate System Grade point average is measured on the vertical axis and study time on the horizontal axis. Albert E., Alfred E., and their classmates are represented by various points. We can see from the graph that students who study more tend to get higher grades. use. 7 Plotting Income and Consumption Data for Households TABLE 1A.2 Consumption Expenditures and Income, 2008 Bottom fifth 2nd fifth 3rd fifth 4th fifth Top fifth Average Income Before Taxes Average Consumption Expenditures $ 10,263 27,442 47,196 74,090 158,652 $ 22,304 31,751 42,659 58,632 97,003 FIGURE 1A.2 Household Consumption and Income A graph is a simple two-dimensional geometric representation of data. This graph displays the data from Table 1A.2. Along the horizontal scale (X-axis), we measure household income. Along the vertical scale (Y-axis), we measure household consumption. Note: At point A, consumption equals $22,304 and income equals $10,263. At point B, consumption equals $31,751 and income equals $27,442. Curves in the coordinate system Negatively related variables ◦ The two variables move in opposite direction ◦ Downward sloping curve Positively related variables ◦ The two variables move in the same direction ◦ Upward sloping curve Movement along a curve Shifts in a curve © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a passwordprotected website for classroom use. 9 Slope slope A measurement that indicates whether the relationship between variables is positive or negative and how much of a response there is in Y (the variable on the vertical axis) when X (the variable on the horizontal axis) changes. Y2 Y1 Y X X 2 X 1 FIGURE 1A.3 A Curve with (a) Positive Slope and (b) Negative Slope A positive slope indicates that increases in X are associated with increases in Y and that decreases in X are associated with decreases in Y. A negative slope indicates the opposite—when X increases, Y decreases; and when X decreases, Y increases. FIGURE 1A.4 Changing Slopes Along Curves Demand Curve Price of Novels $11 10 9 8 7 6 5 4 3 2 1 (5, $10) (9, $9) (13, $8) (17, $7) (21, $6) (25, $5) Demand, D1 30 Quantity of novels purchased The line D1 shows how Emma’s purchases of novels depend on the price of novels when her income is held constant. Because the price and the quantity demanded are negatively related, the demand curve slopes downward. 0 5 10 15 20 25 . 13 Shifting Demand Curves Price of $11 Novels 10 9 8 7 6 5 4 3 2 1 0 When income increases, the (13, $8) demand curve shifts to the right. (16, $8) (10, $8) D2 (income= $40,000) When income decreases, the demand curve shifts to the left. 5 D3 (income= D1 $20,000) (income= $30,000) 10 1315 16 20 25 30 Quantity of novels purchased The location of Emma’s demand curve for novels depends on how much income she earns. The more she earns, the more novels she will purchase at any given price, and the farther to the right her demand curve will lie. Curve D1 represents Emma’s original demand curve when her income is $30,000 per year. If her income rises to $40,000 per year, her demand curve shifts to D2. If her income falls to $20,000 per year, her demand curve shifts to D3. 14 Slope ◦ Ratio of the vertical distance covered ◦ To the horizontal distance covered ◦ As we move along the line Δ (delta) = change in a variable The “rise” (change in y) divided by the “run” (change in x). y Slope x . 15 Slope ◦ Fairly flat upward-sloping line Slope = small positive number ◦ Steep upward-sloping line Slope = large positive number ◦ Downward sloping line Slope = negative number ◦ Horizontal line Slope = zero ◦ Vertical line Infinite slope . 16 Calculating the Slope of a Line Price of Novels $11 10 9 8 7 6 5 4 3 2 1 0 (13, $8) 6-8=-2 (21, $6) 21-13=8 5 10 13 15 Demand, D1 20 21 25 30 Quantity of novels purchased To calculate the slope of the demand curve, we can look at the changes in the x- and y-coordinates as we move from the point (21 novels, $6) to the point (13 novels, $8). The slope of the line is the ratio of the change in the y-coordinate (–2) to the change in the x-coordinate (+8), which equals –1⁄4. 17