The purpose of this exercise is for you to a.) become comfortable with mathematical equations and b.) to recognize how equations appear in graphs by using EXCEL.

First of all, don't get nuts! Mathematics is only a language that describes nature. It is not nature itself. It may please you to know that nature is far too complicated to be fully described by mathematics. This makes some mathematicians crazy. The reason why mathematicians keep on doing math is to try to overcome its clumsiness describing nature. Second, mathematics has probably been taught to you as a language. That is, as a series of abstract rules and procedures (i.e., like French grammar or music notation). It is only at the last step that teachers, if they have the time or experience, suggests its relevance to everyday life. In this course we approach mathematics another way. First, here are the aspects of nature and, second, here is how we use the mathematics to describe nature. In other words, you don't fix a car by understanding the physics of movement. You fix it by recognizing each part and knowing what it does. You don't need to play music by knowing musical notation. You can play it directly by using your ears, mind and fingers.

Mathematical functions describe relationships between two variables, called X and Y. The value of X determines the values of Y. For example, the more money you have (i.e., money is X), the more you can buy (i.e., what you buy is Y).

What follows you really don't need to know for this exercise. It is here to clear up terminology you will or have probably encountered. It's good for you. The X variable is called the independent variable because that's what you start out with. It's given like the number of days in a week. The Y is called the dependent variable because the value of X when pushed through a mathematical equation gives you the value of Y.

Functions can be direct or indirect. In a direct relationship, the greater the X value, the greater the Y value (i.e., the more money you have, the more you can buy). In a indirect relationship, the greater the X value, the smaller the Y value (i.e., the more you run, the less energy you have.).

As X increases, Y will either increase or decrease. (It may also stay the same, go from one to the other, or be weird but we don't need to know that here.) For the moment let's consider that as X increases, Y increases. Now, as we go from X equals 0 to 1 to 2 to 3, etc., Y may also increase at the same rate from 0 to 1 to 2 to 3, etc. But is could increase at a different rate. That is as X goes from 0 to 1 to 2 to 3, etc., Y goes from 0 to 2 to 4 to 6, etc. Or even worse, Y goes to 0 to 1 to 4 to 9, etc.

A mathematical equation is the relationship, also between X and Y. But equations can be lumped together, or classified, by the way Y increases as X increases. Each of these types is referred to as a function. The kinds of functions we will look at are as follows.

• Linear Functions

• Power Functions

• Exponential Functions

• Logarithmic Functions

• Periodic Functions

• Quadratic Functions

• Combinations of Functions

Now get into EXCEL and we will continue.

Linear functions result in a straight line in an XY plot.

Our EXCEL sheet will consist of a series of columns of X and Y. But the Y will be determined by calculating various functions using EXCEL. The actions you will need to do on your EXCEL sheet are indicated by a number at the extreme left like the one on the next line. The sentences without such numbers provide background to help you know what you're doing.

1. Type your name in A1, the date in A2, and Mathematical Functions in A3.
2. Type Direct Linear Function in A5, m= 1 on A6, and m= 5 on C6.
3. Type Indirect Linear Function in E5 and type m= -1 in E6.
4. Type X in A7, Y in B7, C7 and D7.
5. Type the numbers 0,1,2,3,4,5,6,7,8, and 9 in boxes A8 through A17.

Remember the formula for a line Y = mX + b from high school. It is a generalized formula for a linear function. The letter m is the slope of the line and b is where the line intercepts the Y-axis. If you didn't get it in high school or don't remember it, here it is anew. The slope m is critical. If m is a positive value, the function is direct (i.e., as X increases, Y increases). If m is a negative value, the function is indirect (i.e., as X increases, Y decreases). We will simplify matters by saying b equals zero, thus the equation becomes Y = mX.

6. Go to B8 and type =A8. The same value as A8 should appear in B8. This is the value of Y when X = 0 and the slope is 1. Now click on copy in B8. Highlight B8 through B17 and click on paste. You now have all the values of Y next to their corresponding X's.
7. Go to C8 and type=5*A8. We here you have a slope (or m) of 5 and have to multiply the X value by it. You will get an Y equal to zero. As before, click on copy in C8, highlight C8 through C17, and click on paste.
8. Go to D8 and type =-1*A8. We here have a indirect linear function since its slope is -1. As before, click on copy in D8, highlight D8 through D17, and click on paste.

Now we're ready to make some charts.

9. Highlight A7 through D17 and click on the chart wizard. Click on XY (Scatter), click on lower left-hand box, and click on next. Now the second window will show your graph. Click Next.
10. This window allows you to enter text onto your graph. Type Linear Functions (m=1,5,-1) in Title Box. Also add you initials to the title in order to recognize your work when it comes out of the printer. Then type in X in the X box and Y in the Y box. Click Next.
11. This window allows you to enter your graph as a separate sheet or as part of your data. Click on as a separate sheet.
12. Print this sheet.

Now we are ready for the other functions.

Power & Exponential Functions

Power and exponential functions are non-linear and may be direct or indirect. They appear as a curve on the XY Plot. If it is a direct function, as X increases, Y still increases. However, the increase in value from one Y to the next gets larger and larger. If it’s an indirect function, as X increases, Y decreases. However the decrease in the value from one Y to the next gets smaller and smaller. The difference between a power function and an exponential function has to do with the value of the exponent, indicated by the letter a in the formula below. A power function is the simplest kind of exponential function.

The formula for a power function is Y = mx^a. The exponent is indicated by the superscript a. This formula could have a Y intercept but let's assume it equals zero. Let's also assume that the slope m equals 1. Now the equation becomes Y = x^a. A power function has only one value in its exponent. An exponential function has more than one value in its exponent. A simplified formula for an exponential function could be Y = x^ab. Again assume a zero value for the Y intercept and a 1 value for the slope.

Now we will make a chart for a direct power function (Y = mx^a) and a indirect power function (Y = mx^-a). The actual formulas for the power functions we are calculating is Y = mx² for the direct function and Y = mx^-2 for the indirect power function.

13. Type Direct Power Function in F5 and exp 2 in F6.
14. Type Indirect Power Function in G5 and exp-2 in G6.
15. Type X in E7, Y in F7 and G7.
16. Type the numbers 0,1,2,3,4,5,6,7,8, and 9 in boxes E8 through E17.
17. Type =E8^2 in F8. Then copy F8, highlight F8 through F17, and paste.
18. Type in =E8^-2 in G8. Then copy G8, highlight G8 through G17, and paste. [Notice that you get a non-numerical value when X=0. This is appropriate. If you get bogus values or non-numerical values for G8 through G17, delete G8 and go to G9. Then type =E9^-2, then copy G9, highlight G9 through G17 and paste.]
19. Highlight E7 through G17 and click on chart wizard.
20. Repeat steps to make a XY scatter plot but this time for the two power functions. Type Power Functions (a=2,-2) in Title Box. Also add you initials to the title in order to recognize your work when it comes out of the printer.

Now it's on to the exponential function.

21. Type exponential function in I5 and exp ab in I6.
22. Type a in I7, b in J7, X in K7 and Y in L7.
23. Fill in the values 0,1,2,3,4,5,6,7,8,9 in I8 through I17. Fill in the same sequence in J8 through J17 and K8 through K17. This means the exponents a and b as well as the value for X will go from O to 9.
24. Go to L8 and type =I8^(J8*K8). Then copy L8, highlight L8 through L17, and paste. The value of Y will be non-numerical when X, a, and b=0. Delete L8. Go to L9 and type =I9^(J9*K9). Copy L9, highlight L9 through L17, and paste.
25. Now highlight K7 through L17 and use your chart wizard as before. This time the title should be Exponential Function and your initials.

Now we are ready for logarithmic functions. Observe the correct spelling for loga-rith-mic.

 

 

Semi-Log and Log-Log Functions

There are two kinds of logarithmic functions. If Y =log(X), it is a semi-logarithmic function. If Log(Y) = Log(X), it is a log-log logarithmic function. Let's set up the charts for these functions. These functions are non-linear and appear as curves on regular graph paper. Regular graph paper is made up of squares. The distance between the vertical lines and the horizontal lines are the same. However, semilog and log-log graph paper is made up of rectangles not squares. The distance between the vertical lines, or the horizontal lines, in the case of semilog graph paper or both in the case of log-log graph paper get closer and closer together as you go from 1 to 10. The use of semilog or log-log graph paper turns curves from regular graph paper into linear functions. It is important to recognize that you cannot for to zero or negative numbers using logarithmic functions.

The graphs on page 421 of your text are semilogarithmic functions plotted on semilog graph paper. Notice that the function itself looks linear. The Y values are logarithmic. That is, as you go up the Y-axis in graph (a), you pass from values in the one's (i.e., 1 through 9) to values in the ten's (i.e., 10 through 90) to values in the 100's (i.e., 100 to 900), and to values in the 1000's (i.e., 1000 to 9000). Also the numbers within one order of magnitude (i.e., ten's) get closer together as you pass from 10 to 90 for instance. However the values on the X-axis have vertical lines that are equally spaced from one another.

The graph on page 397 of your text is a log-log function on log-log graph paper. The Y-axis values go from ones to tens to hundreds to thousands, to ten thousands to hundred thousands and the lines get closer and closer as you pass from 1 to 9. The values along the x-axis pass from the hundredths to tenths to ones to tens and the lines get closer and closer as you go from 1 to 9.

Exponential and power functions can be converted into logarithmic functions. You can see an example of this on page 391. Notice the box in the graph where the power function S = 178Q^1.75 is transformed into the logarithmic function log S = 2.25 + 1.75 log Q. [The value 2.25 is the Y intercept of the linear function while the power 1.75 of the power function has been transformed into a slope of the same value in the logarithmic function.]. Finally, when power or exponential functions, which appear as curves on regular graph paper, are plotted on log-log or semilog graph paper, the line plotted becomes linear. This is shown on page 481 of your text where isostatic rebound (i.e., the expansion of the earth's surface upward once glacial ice is removed) is a curve in the uppermost graph plotted on regular graph paper and a straight line in the middle graph plotted on semi-logarithmic paper.

Now back to EXCEL.

26. Go to A20 and type logarithmic functions. Type semi-logarithmic function in A21 and log-logarithmic function in D21.
27. Type X in A22, Y in B22, X in D22, Log X in E22, and Log Y in F22.
28. Fill in the values from 0 through 9 in A23 through A32 and in D23 through D32.
29. Go to B23 and type =log(A23). You will get a non-numerical value. So delete it and go to B24 and type =log(A24). Now copy B24, highlight B24 through B32, and paste.
30. Highlight A22 through B32. Make a XY plot as before with this title being Semi-Logarithmic Function.
31. But we also have to change the name of the X-axis to Log X. So type in Log of X in the slot allotted.
32. Once the graph is finished, we have to alter the graph scale for X. To do this, first make sure the boxes on the edges and middle of lines are present. If they are not, single click in the white parts of the graph. That will bring the boxes up. Now double click on a number in the X scale. A window will come up with several tabs along the top of the window. Click on scale. There is a logarithmic scale box. Click the box once and an x will appear in the box. Now click OK. You will see that the X scale now goes from 0.1 to 1. You cannot get a zero or negative number for a log number. Print your chart.
33. Now go to E23 and type =log(D23). You guessed it! Another non-numerical value. So delete it and go to E24 and type =log(D24). Copy E24, highlight E24 through E32, and paste.
34. Since the log of X is equal to the log of Y, go to F24 and type =E24. Copy F24, highlight F24 through F32, ands paste.
35. Highlight E22 through F32 and go to the chart wizard as before. The title for this should be Log-logarithmic Function. The X should be Log X and the Y should be Log Y.
36. Once you've got the finished chart, you will have to change the scales of both X and Y. So double click on and X value. Find and click on the logarithmic box in the scale window and click OK. Do the same for the Y scale. Again you will get values from 0.1 to 1 on the X and Y scales.

Logarithmic functions are initially curves in an XY plot. But when you change the scales to logarithmic scales, the graphs give you a straight line. But notice the spacing between one value and the next gets smaller and smaller toward the higher values. This is characteristic of logarithmic functions.

 

 

Quadratic Functions

Remember in high school algebra when you had to multiply (X + 1) by (X + 1). You'd get a product of X² + 2X + 1. This is a quadratic equation. X appears several times on the right side of the equal sign. This produces another series of curves on an XY plot. So here we go again.

37. Go to G20 and type Quadratic Function
38. Go to G22 and type X and H22 and type Y.
39. Type in 0,1,2,3,4,5,6,7,8,9 in G23 through G32.
40. Go to H23 and type =(G23^2)+(2*G23)+1. This is the formula for X² + 2X + 1. Then Copy H23, highlight H23 through H32, and paste.
41. Again make XY plot by first highlighting G22 through H32. The title this time in Quadratic function.

Finally, periodic functions.

 

Periodic Functions

Tides and seasons are examples of periodic functions. The tide rises, crests, declines, bottoms out, and rises again. Another words, while X increases, Y increases then decreases or decreases then increases.

42. Go to J20 and type Periodic Function. Periodic Functions include sins, cosines, tangents, etc. in their formulae.
43. Go to J22 and type X and at K22 type Y.
44. Type 0,1,2,3,4,5,6,7,8,9 in J23 through J32.
45. Go to K23 and type =sin(J23).
46. Copy K23, highlight K23 through K32, and paste.
47. Make your last chart (yipee!!!) but this time use the title Periodic Function.
48. Print charts.

Quite often, you will be a graph were the slope of the line of a linear function changes. For example, it may pass from an indirect function to a direct function as in figure 3.7 of your text. It will go from a shallow slope to a steep slope. Or even worse, it will go from a linear function to a non-linear function or to a periodic function. Some examples of combined functions are seen in figure 20.6 of your text. In nature, the change in function reflects the fact that more than a single independent variable controls the dependant variable. For example, the amount of evaporation off of a lake's surface could be controlled by temperature, humidity and wind. If it were just a function of temperature it would look like figure 4.7 in your text, a simple power function. But if temperature rises, then a little later the air gets drier and still later the wind picks up, the curve would suddenly steepens with each additional independent variable. This curve would be the result of a combination of functions. The good news is that you don't have to make this kind of function today.

Now close your eyes and let your mind relax. Once the hyperness of the previous activities wears off, open your eyes and look at your work-beautiful XY plots. Spread all the charts out in front of you and see how their curves or lines (in the case of linear and logarithmic functions) differ. Use these charts to answer the following questions.

1. What part of the formula for a line indicates whether the independent variable (i.e., X) radically or slightly changes the value of the dependent variable (i.e., Y)?
2. What part of the formula for an exponential function indicates whether the independent variable (i.e., X) radically or slightly changes the value of the dependent variable (i.e., Y)?
3. What part of the formula for a line indicates whether the relationship is direct or indirect?
4. What part of the formula for a power function indicates whether the relationship is direct or indirect?

And you thought you couldn't do mathematics!!!