1. Class 7.2: Graphical Analysis and Excel Solving Problems Using Graphical Analysis

2. Learning Objectives Learn to use tables and graphs as problem solving tools Learn and apply different types of graphs and scales Prepare graphs in Excel Be able to edit graphs Be able to plot data on log scale Be able to determine the best-fit equations for linear, exponential and power functions

3. Exercise Enter the following table in Excel You can make your tables look nice by formatting text and borders

4. Axis Formats (Scales) There are three common axis formats: Rectilinear: Two linear axes Semi-log: one log axis Log-log: two log axes Length (km) 1 km10 km Log scale: 1 km Length (km) Linear scale:

5. Use of Logarithmic Scales A logarithmic scale is normally used to plot numbers that span many orders of magnitude

6. Creating Log Scales in Excel Exercise (2 min): Create a graph using x and y1 only.

7. Creating Log Scales in Excel Now modify the graph so the data is plotted as semi-log y This means that the y-axis is log scale and the x-axis is linear. Right click on the axis to be modified and select “format axis”

8. Creating Log Scales in Excel On the Scale tab, select logarithmic “OK” Next, go to Chart Options and select the Gridlines tab. Turn on (check) the Minor gridlines for the y-axis. “OK”

9. Result: Graph is straight line.

10. Exercise (8 min) Copy and Paste the graph twice. Modify one of the new graphs to be semi-log x Modify the other new graph to be log- log Note how the scale affects the shape of the curve.

11. Result:semi-log x

12. Result: log-log New Graph 1 10 100 1000 10000 110100100010000 x y1

13. Equations The equation that represents a straight line on each type of scale is: Linear (rectilinear): y = mx + b Exponential (semi-log): y = be mx or y = b10 mx Power (log-log): y = bx m The values of m and b can be determined if the coordinates of 2 points on THE BEST-FIT LINE are known: Insert the values of x and y for each point in the equation (2 equations) Solve for m and b (2 unknowns)

14. Equations (CAUTION) The values of m and b can be determined if the coordinates of 2 points on THE BEST- FIT LINE are known. You must select the points FROM THE LINE to compute m and b. In general, this will not be a data point from the data set. The exception - if the data point lies on the best- fit line.

15. Consider the data set: XY 1 4 2 8 310 412 511 616 718 819 920 1024

16. Team Exercise (3 minutes) Using only the data from the table, determine the equation of the line that best fits the data. When your team has completed this exercise, have one member write it on the board. How well do the equations agree from each team? Could you obtain a better “fit” if the data were graphed?

17. Which data points should be used to determine the equation of this best-fit line?

19. Comparing Results How does this equation compare with those written on the board (i.e- computed without graphing) ? CONCLUSION: NEVER try to fit a curve (line) to data without graphing or using a mathematical solution ( i.e – regression)

20. What about semi-log graphs? Remember, straight lines on semi-log graphs are EXPONENTIAL functions.

21. What about log-log graphs? Remember, straight lines on log-log graphs are POWER functions.

22. Example Points (0.1, 2) and (6, 20) are taken from a straight line on a rectilinear graph. Find the equation of the line, that is use these two points to solve for m and b. Solution: 2 = m(0.1) + ba) 20 = m(6) + bb) Solving a) & b) simultaneously: m = 3.05, b = 1.69 Thus: y = 3.05x + 1.69

23. Pairs Exercise (10 min) FRONT PAIR: Points (0.1, 2) and (6, 20) are taken from a straight line on a log-log graph. Find the equation of the line, ie - solve for m and b. BACK PAIR: Points (0.1, 2) and (6, 20) are taken from a straight line on a semi-log graph. Find the equation(s) of the line, ie - solve for m and b.

24. Interpolation Interpolation is the process of estimating a value for a point that lies on a curve between known data points Linear interpolation assumes a straight line between the known data points One Method: Select the two points with known coordinates Determine the equation of the line that passes through the two points Insert the X value of the desired point in the equation and calculate the Y value

25. Individual Exercise (5 min) Given the following set of points, find y2 using linear interpolation. (x1,y1) = (1,18) (x2,y2) = (2.4,y2) (x3,y3) = (4,35)

26. Assignment #13 DUE: TEAM ASSIGNMENT See Handout