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Jerry Crawford – June 23, 2011 1 Thinking Ahead about Graphing – You should understand these questions fully before the next class. Check your answers with the key on your instructor’s website. You can get help with this work from the following sources: • • • Visit your instructor during office hours Go to the MAC (Math Assistance Center) 700 BH Go to http://www.themathpage.com/aPreCalc/graphs-of-functions.htm Experimental data is often graphed in order to see if any type of relationship exists between the independent variable and the dependent variable. If the plot shows random behavior, perhaps there is no useful relationship. Even then, it is often possible to see some range of expected behavior for a particular variable. For example, Graph #1 shows that the average height of adults is not particularly related to their month of birth, but a trend shown by the graph is that the average adult height falls mostly in the range between 5 feet and 6.5 feet. Graph #1 Three particular relationships between independent and dependent variables are fairly common in the physical sciences. The first is a linear relationship,= y mx + b , shown in Graph #2, which is a plot of the data from Table 1, the speed of fall (dependent variable) versus time of fall (independent variable). This graph shows a representation of the data as if handplotted on linear graph paper. Table 1: Speed of Fall vs. Time of Fall. Time (s) Speed (m/s) 0 0 1 9.8 2 19.6 3 29.4 4 39.2 5 49.0 6 58.8 Graph #2: Linear plot of Speed of Fall vs. Time The next common relationship is a power law relationship, y = Ax , where the dependent variable is related to some power of the independent variable. If data related in this way are plotted on linear paper, a curved line results unless the power, m = 1, in which case the data are linearly related as in the first example. It is possible to “straighten out” power law data by m Jerry Crawford – June 23, 2011 2 plotting on log-log paper. This has the same effect as plotting the logarithm of y against the logarithm of x on linear paper. This is demonstrated in the log-log plot of Graph #3. If the logarithm is taken of both sides of the power law relation, the result is: log( = y ) log( A) + m log( x) . This is equivalent to a linear relationship with log(y) being the dependent variable, log(x) the independent variable, m the slope of the straight line, and log(A) the intercept. Table 2: Speed of Fall vs. Distance of Fall Distance (m) Speed (m/s) 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 4.4 6.3 7.7 8.9 9.9 10.8 11.7 12.5 13.3 14.0 19.8 24.2 28.0 31.3 34.3 37.0 39.6 42.0 Graph #3: Log-Log Plot of Speed of Fall vs. Distance of Fall Note the both the horizontal and vertical scales are non-linear, and that after reaching 10 (1,2,3,…8,9,10), the scale intervals increment by 10’s (10,20,30,….80,90,100) instead by 1’s. Exercise: Plot the speed versus distance data on a graph with linear scales, either by hand, or with a calculator or computer (if you are familiar with using those to graph), to see that the data relationship is not linear. The speed actually varies as the square root of the distance. The third common relationship between measured variables is exponential, y = Ae , where x is the independent variable and y the dependent. An example of this is radioactive decay. If the natural logarithm of both side of this equation is taken, the result is: mx ln( y ) = ln( A) + mx ln(e) = ln( A) + mx . Jerry Crawford – June 23, 2011 3 This is equivalent to a linear relationship with ln(y) being the dependent variable, x the independent variable, m the slope of the straight line, and ln(A) the intercept. If the natural logarithm of y is plotted versus x on linear paper, a straight line should result. Alternatively, y may be plotted against x on semi-logarithmic paper, and a straight line will result. The vertical axis is logarithmic and the horizontal axis linear for semi-log plots. Table 3: Nuclei Remaining vs. Time 12000 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 10000 9512 8607 7408 6065 4724 3499 2466 1653 1054 639 369 202 106 52 25 11 5 2 1 0 Graph #4 shows the curve resulting from plotting the data from Table 3 on a linear graph. Graph #5 shows the same data plotted on a semi-log graph, resulting in a straight line. Note that the vertical scale is logarithmic, with powers of ten sharing equal increments of height. 10000 Number of Nuclei Remaining Nuclei 8000 6000 4000 2000 0 0 20 40 60 80 100 120 Time (days) Graph #4: Linear plot of Number of Nuclei vs. Time 10000 1000 Number of Nuclei Time (days) 100 10 1 0 20 40 60 80 100 120 Time (days) Graphs #4 and #5 were produced using Microsoft Excel, taking Graph #5: Semi-log plot of Number of Nuclei vs. Time advantage of the feature allowing the axis scales to be changed from linear to logarithmic. The first graph was plotted, and then copied, and the vertical axis was re-scaled to be logarithmic. Jerry Crawford – June 23, 2011 The ability to quickly re-scale is common in most graphing software, and is a very useful timesaving feature. Exercise: Plot the data in the following table on linear, log-log, and semi-log graphs either by hand, or using calculator or computer graphing software, to determine if the relationship is linear, a power law, exponential, or none of these. Graph the thickness of the absorber as the xaxis variable (independent) and the relative intensity as the y-axis variable (dependent). Graphing paper for hand-graphing is available for free online at http://www.intmath.com/downloads/graph-paper.php. Thickness (cm) 0.1 0.5 1 1.5 2 3 5 7.5 10 Relative Intensity 0.917 0.65 0.422 0.274 0.178 0.075 0.013 0.0016 0.00018 4