## 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

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.)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!)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.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}

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?

Food for thought: your phone/computer do this 60–120 times

**per second**whenever the display is on! ↩︎