Activity 1: Drawing Plots by Hand

Learning Objectives

• Students will understand the mechanics of plotting functions of the form $z=f(x,y)$
• Students will get practice using tables and plots simultaneously.
• Students will get practice evaulating functions of two variables.

Instructions

1. Open Excel and create a new spreadsheet. Label the top row starting in cell B1 with the values $-3,-2,\dots ,2,3.$ Label the leftmost column starting at cell A2 with the same values. The column labels will correspond to x-values, the row labels will correspond with y-values. (You may want to label your y-values in decreasing order to make your graph resemble the typical xy-axes.)

2. We are now going to populate our table with the function values $f(x,y)=-1+8e^{\frac{-x^2-y^2}{9}}$.

   In cell B2, enter the formula =-1+8*EXP((-1/9)*(B$1^2+$A2^2)). Copy this across each cell in your table. (Excel should make the associated changes as you drag the formula across.) (If you’re having issues, make sure there’s not a space before the = sign!)

3. Select cells B2 through H8. Then, click Home > Conditional Formatting > Color Scales. Click any of the color scales. This creates a “heat map” of our function. Larger values get colored with one color, smaller values get colored with another.

4. This essentially is a 7x7 pixel plot of the function $f(x,y)$. In small groups, answer the following questions:

   • Suppose we wanted to increase the resolution of our image by a factor of two in both the $x$ and the $y$ directions. so that the spacings are $\delta x=\delta y=0.5$ instead of $1$. How many pixels would your new image use? Compare this with how many new calculations you would have to do to draw the trace $z=f(x,0)$ (i.e. the $y=0$ trace).
   • Compare and contrast graphing functions of one variable and two variables. What takes more work? What produces cooler graphs? Why does one take more work than the other?
   • Now, imagine you need to build an “image” with a resolution of 1920x1080px. (aka “Full HD” or “1080p”) How many pixels are contained in this image? Do you think it is feasible to do this “by hand”, using our method?¹

5. Use one or more of the plotting tools on the Resources Page to make a 3D plot of our function $f(x,y)=8e^{\frac{-x^2-y^2}{9}}$. How do the features of your plot compare to the features of the the 3D plot?

Activity Solution