Introduction to Mathematica for High School Math (for Students and Teachers) #8
Author
Ruth Dover
Title
Introduction to Mathematica for High School Math (for Students and Teachers) #8
Description
Introduction to Mathematica for High School Math
Category
Educational Materials
Keywords
Mathematics, education
URL
http://www.notebookarchive.org/2021-09-6h297x2/
DOI
https://notebookarchive.org/2021-09-6h297x2
Date Added
2021-09-14
Date Last Modified
2021-09-14
File Size
21.26 kilobytes
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Rights
Redistribution rights reserved
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Tutorial 8: More 2D Graphing
Tutorial 8: More 2D Graphing
R. Dover, IMSA
Parametrics
Parametrics
For the command , you need to enter the functions , as a list of functions. Beyond that, the coding and the options work in a manner similar to those for :
ParametricPlot
x=f(t)
y=g(t)
Plot
ParametricPlot[{Cos[t],Sin[t]},{t,0,2π}]
Check to see what options are available with ParametricPlot:
Options[ParametricPlot]
Then create a couple of graphs of your own.
Polar Graphs
Polar Graphs
These examples should be clear:
PolarPlot[3Sin[6θ],{θ,0,2π}]
PolarPlot3Sin,{θ,0,4π},PlotStyle{Green,Thick}
7θ
2
Implicit Graphs
Implicit Graphs
For this, we use , which is related to 3D graphs. It is necessary to give both and values for the domain. (We will stick with the 2D approach for now.) For these graphs, using will often be helpful, along with showing the axes:
ContourPlot
x
y
AspectRatio
ContourPlot+1,{x,-4,4},{y,-3,3},AspectRatioAutomatic,AxesTrue
2
x
9
2
y
4
There are many very interesting implicitly defined graphs:
ContourPlot[Sin[3x]+Cos[2y]+x+y1/2,{x,-4,4},{y,-4,4}]
So What about 3x?
So What about ?
3
x
Check out the following:
3
-8
Plot[,{x,-8,8}]
1/3
x
Probably not what you wanted. The Wolfram Language™ works in complex mode, so the cube root of a negative number offers many complex solutions—which are not plotted. Here are two options, one specific and one that can be generalized for other roots:
Plot[CubeRoot[x],{x,-8,8}]
Plot[Surd[x,3],{x,-8,8}]
More Help
More Help
As we have seen, there is lots and lots of help available in the Wolfram Language. In fact, it can be somewhat overwhelming before you have some basis with the software. With a sense of the fundamental structures of the software, it becomes much easier to use the Documentation Center. One more time, from the Help menu, choose Wolfram Documentation. Again, find Visualization and Graphics, and then choose Function Visualization. Check out more of the possibilities for ParametricPlot or or . (We will get to the 3D stuff soon enough.)
ContourPlot
PolarPlot
And as yet another quick option, typing a “?” and a command will give you some basic clues:
?Plot
Exercises
Exercises
(1) Define both and and plot the parametric curve for . Make the graph thick and green. (This is a Lissajous curve.)
x(t)=cos(3t)
y(t)=2sin(5t)
(x(t),y(t))
0≤t≤2π
(2) Plot the polar curve on the interval .
r(θ)=θsin(3θ)
0≤θ≤2π
(3) Plot the graph of the implicit curve -3+4y=0. As a bonus, replace the on the right with and animate the graph for . Note that as you move the slider, Mathematica® draws very rough graphs in order to keep up with the slider. Some of those really do not look very good, including many straight segments, but as soon as you release the slider for a specific value of , Mathematica will leave a good graph for you to see.
3
x
2
x
3
y
0
k
−6≤k≤6
k
(4) What do you think the Wolfram Language will do when you try to plot ? Fix this to get a complete graph.
y=
2/3
x
Cite this as: Ruth Dover, "Introduction to Mathematica for High School Math (for Students and Teachers) #8" from the Notebook Archive (2021), https://notebookarchive.org/2021-09-6h297x2
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