Educational Materials

Introduction to Mathematica for High School Math (for Students and Teachers) #5

Ruth Dover

Author

Ruth Dover

Title

Introduction to Mathematica for High School Math (for Students and Teachers) #5

Description

Introduction to Mathematica for High School Math

Tutorial 5: Plotting

R. Dover, IMSA

Basic Graphs

Graphics in Mathematica® are amazing! This tutorial will help you get started.

The code following is for the graph of

f(x)=xsin(2x)

for

−1≤x≤5

(again, note the space between

and

Sin[2x]

to denote multiplication. Otherwise, this would be read as one variable,

xSin

, with a four-letter name):

Plot[xSin[2x],{x,-1,5}]

The range is chosen automatically to show the most important features of the graph, but this may be controlled by the user, along with many other options. Several examples are given. Look at each one carefully. The function is given first, followed by the curly braces with the domain. After that, options may be given in any order. There are lots of arrows, braces, capitals and commas to watch. Understanding the system will make it much easier to include a variety of options:

Plot[xSin[2x],{x,-1,5},PlotRange5]

Plot[xSin[2x],{x,-1,5},PlotRange{-3,8}]

Plot[xSin[2x],{x,-1,5},PlotStyleRed]

Dashed

and

Thick

are given default values, but these may be controlled:

Plot[xSin[2x],{x,-1,5},PlotStyle{Dashed,Thick}]

Plot[xSin[2x],{x,-1,5},PlotStyle{Dashing[{.05}],Thickness[.03]}]

Plot[xSin[2x],{x,-1,5},AspectRatioAutomatic,Ticks{{π/2,π,3π/2},Automatic}]

Plot[xSin[2x],{x,-1,5},PlotLabel"ƒ(x) = x sin(2x)"]

Create one of your own graphs. Then make several changes. Experiment a little.

Help!

Clearly, there are lots of commands and options in the Wolfram Language™. And clearly, it is impossible to remember all of them. Fortunately, Mathematica provides a lot of help. You may have already seen or tried this, but in a new cell, start to type “Plot” by typing just “Pl”. Pause. Two letters will allow you to see all commands that begin with these letters. Clicking the box with two downward arrows will show more commands.

Now, choose

Plot

from the list. Click the two downward arrows again. This time, you will see the basic syntax. Choose the first function (for one graph), and the template will appear. Nice!

This can obviously be quite helpful. Try typing just the first two letters of a few other commands.

But what about available options?

Options[Plot]

Whoa! Read through that list carefully. In each case, the default value for each option is shown.

More Graphs

To graph two or more functions on one graph, create a list of functions:

Plot[{xSin[2x],

-3x+1},{x,-4,4}]

Note that by default, Mathematica shows the graphs in two different, muted colors. Sometimes, this is fine, but you may also want to control more:

Plot[{xSin[2x],

-3x+1},{x,-4,4},PlotStyle{Red,Blue}]

To add more styles to each graph, we need a list of lists:

Plot[{xSin[2x],

-3x+1},{x,-4,4},PlotStyle{{Red,Thick},{Blue,Dashed,Thick}}]

As stated before, you may control the thickness and the size of the dashes. But sometimes, simply writing

Thick

and

Dashed

is good enough!

Plot[{xSin[2x],

-3x+1},{x,-4,4},PlotStyle{{Red,Thickness[.003]},{Blue,Dashing[{.05,.02}],Thickness[.01]}}]

Colors

Some colors such as

Red

and

Blue

have been used, and the Wolfram Language™ understands these and others. (To see what Mathematica knows, from the Palettes menu, choose ColorSchemes. Under the section Named, look at the colors under System.)

Colors may be chosen in other ways. RGB stands for Red, Green, Blue and requires three numbers all from 0 to 1, inclusive. Try some different numbers:

Plot[xCos[2x],{x,0,2π},PlotStyle{Thick,RGBColor[1,0,0]}]

Another option is to use

Hue[k]

, where

is a number between 0 and 1. Try other values of

, remembering your rainbow:

Plot[xCos[2x],{x,0,2π},PlotStyle{Thick,Hue[0]}]

Check out more possibilities in the palette if you wish.

Asymptotes

Look at the basic

Plot

following. In recent versions, the Wolfram Language has figured out how to avoid showing the vertical asymptotes that many other programs show:

In[]:=

g[x_]:=

(x-2)(x+1)

In[]:=

Plot[g[x],{x,-3,4}]

If you like asymptotes and want to see something a little more complicated, check this out:

In[]:=

Plot[g[x],{x,-3,4},Epilog{{Thick,Red,Dashed,InfiniteLine[{{-1,0},{-1,1}}]},{Thick,Red,Dashed,InfiniteLine[{{2,0},{2,1}}]}}]

The command

InfiniteLine

draws a line through two given points to the edges of the window.

Two More Options (Optional)

Mesh

Now look at a basic graph with the option

Mesh

Plot[Sin[x],{x,0,2π},MeshAll]

What is this doing? It is showing you all of the points that it is plotting to make the graph. The remainder of the graph is created by joining these points together with straight line segments. Note that calculators use equally spaced points in all cases, while Mathematica cleverly puts in more points where the function is more curvy.

Mesh may be used to do a number of other things. This next example gives 12 evenly spaced points within the domain:

Plot[Sin[x],{x,0,2π},Mesh12,MeshStylePointSize[Large]]

For points spaced evenly within the range, use a "pure" function, shown at the end:

Plot[



,{x,0,2π},Mesh8,MeshStyle{Red,PointSize[Medium]},MeshFunctions{#2&}]

RegionFunction

This option will allow you to plot certain parts of your function, subject to your restrictions:

Plot[Sin[x],{x,0,2π},RegionFunctionFunction[{x,y},Abs[y]>.3]]

Plot[Sin[x],{x,0,2π},RegionFunctionFunction[{x,y},.3<Abs[y]<.8]]

Palettes (Again)

In the palette, under Basic Commands, find the button 2D. Look around a little. And again, we will come back to some of these other graphs in more detail later. For right now, choose Plot. Fill in a function of your choice, along with the variable, min and max values. Now choose a few of the options and fill in appropriate items. Note the color palette a little further down. Just click a color when needed.

By Options, it says Automatic Positioning. Be sure you noticed what this implies. (It is very nice!)

Exercise

Let

f(x)=cos(x)

and

g(x)=1−

. Define both of these functions. Then plot them together, adding some colors and thickness or dashing to each graph. Put a frame around the plot and label it Approximations.

Be sure to clear

and

when you are done.

Cite this as: Ruth Dover, "Introduction to Mathematica for High School Math (for Students and Teachers) #5" from the Notebook Archive (2021), https://notebookarchive.org/2021-09-6h22zw6