Graphing Conic Sections

Relating changes in equations to changes in graphs

by Jim Mann, Sheldon H.S.

Mathematica initializations

Off[General::"spell1"]; Off[Plot::"plnr"]; Needs["Graphics`Colors`"];

Note to Instructors

One of the biggest problems I have encountered while teaching conic sections is visualizing the effect that changing the equation of a conic section has on its graph.  The purpose and advantage of the following activity is that it utilizes the animation capabilities of Mathematica to show a series of graphs.  In each of these animations, one value from the general form of the equation is changed over a sequence of values.  I hope this will allow students to see what occurs in the graph when the equation is altered.

Overview for Students

While you may or may not have worked with conic sections in the past, this notebook is intended to give you a working idea concerning three conic sections:  circle, parabola, and ellipse.  In each section you will execute a Mathematica input cell and animate it to view a series of graphs.  You will then be asked some questions about what you observed, and how it relates to the general equations.  Do not concern yourself with the Mathematica code.  Your job (in terms of the software) is only to activate the cell (highlight it and push enter) and animate the graphics (highlight the second cell from the left, push apple-y, then push 2 to slow the aciton).  Your brain activity should be focused on observing the movie, finding the relationship between the graph and function (shown above the graph), and answering the questions.

Circles

Circles - Part 1

Here is the first Mathematica command for you to execute.  Remember, highlight the input cell, then push ENTER.

Do[Show[ParametricPlot[{c Sin[t], c Cos[t]},{t, 0, 2 π},PlotStyle -> Thickness[0.005],PlotRange -> {{-5, 5}, {-5, 5}},AspectRatio -> Automatic,PlotLabel->"x^2 + y^2= (\!\(c\))^2",DisplayFunction -> Identity],DisplayFunction -> $DisplayFunction],{c, 1, 5, 0.5}];

Circles - Part 2

Now that you have your feet wet, this process should become a bit easier.  Here is your input cell for the second change we will make in the circle.  Remember, don't worry about the code.  Execute it, animate it, then your purpose in life at that point becomes the description and accounts of the events about to take place.

Do[Show[ParametricPlot[{2 Sin[t] + c, 2 Cos[t]},{t, 0, 2 π},PlotStyle -> Thickness[0.005],PlotRange -> {{-5, 5}, {-5, 5}},AspectRatio -> Automatic,PlotLabel->"(x - \!\(c\))^2 + y^2= 4",DisplayFunction -> Identity],DisplayFunction -> $DisplayFunction],{c, -2, 2, 0.5}];

While watching this episode of "As the Circle Turns," please ponder these questions, and record your answers:

1) Desribe the changes taking place in the graph.
2) In the general form of the equation, changes in what value are   
          being represented.
3) How would you determine this change if you were only given 
          an equation of some circle?

Circle - Part 3

Now we move on to the third and final episode of the circle saga.  At this point, you should be able to anticipate what is going to happen in this animation (this is a good thing).  So, enjoy the show, and be prepared to put everything together into one generalized package.

Do[Show[ParametricPlot[{2 Sin[t], 2 Cos[t] + c},{t, 0, 2 π},PlotStyle -> Thickness[0.005],PlotRange -> {{-5, 5}, {-5, 5}},AspectRatio -> Automatic,PlotLabel->"x^2 + (y - \!\(c\))^2= 4",DisplayFunction -> Identity],DisplayFunction -> $DisplayFunction],{c, -2, 2, 0.5}];

Just as you predicted (if not, don't worry, that skill will come).  For now, please answer these questions:

1) What does the circle appear to be doing?
2) What value is changing in the general form, causing this to 
          take place?
3) What predictions could you make about the graph, if given the 
          equation?
 AND, FINALLY...  (and maybe most importantly)
 4) How do your answers from these three parts fit together to 
           enable you to make a prediction about the size and 
           location of the graph of a circle, given only its
           equation? (This is your idea of how to graph circles)

Parabolas

Plot[(x^2),{x, -2, 2},PlotStyle -> {RGBColor[0, 0, 1], Thickness[0.005]},PlotRange -> {{-4, 4}, {-4, 4}},AspectRatio -> Automatic,PlotLabel->"y = x^2"];

Parabolas - Part 1

For those who have not gone through the section on circles, here is the basic idea of what you are to do.  Your only concern about the Mathematica input cells is to highlight the cell, push enter, and when the graphs are done, highlight the second cell from the left and push apple-y then 2.  Don't worry about what the input means, you are to concentrate your efforts on the animations.

With that in mind, here is the first set of graphs involving parabolas.  As each graphic is being printed, try to anticipate what the movie will look like.

Do[Show[Plot[(((x - c))^2),{x, -3, 3},PlotStyle 
                    -> {RGBColor[0, 0, 1], Thickness[0.005]},PlotRange -> {{-5, 5}, {-5, 5}},AspectRatio -> Automatic,PlotLabel->"y = (x - \!\(c\))^2",DisplayFunction -> Identity],DisplayFunction -> $DisplayFunction],{c, -2, 2, 0.5}];

Did you anticipate correctly?  If you did, great!!!  If not, keep trying, it will happen.  In the mean time, here are the questions for you to answer.

1) Desribe the changes taking place in the graph.
2) In the general form of the equation a(y-k)=(x-h) \(TraditionalForm\`\^2\), changes  in what value are being represented. 3) How would you determine this change if you were only given  an equation of a parabola?

Parabolas - Part 2

By this time, some of you should be sensing that there is a pattern to the events taking place.  This is good, because there has been a pattern thus far.  The question is not only to discover the pattern but to determine if it continues.  One way to find a pattern is to look at examples, and here is another change in the parabola's graph.

Do[Show[Plot[(x^2) + c,{x, -3, 3},PlotStyle 
                    -> {RGBColor[0, 0, 1], Thickness[0.005]},PlotRange -> {{-6, 6}, {-6, 6}},AspectRatio -> Automatic,PlotLabel->"(y - \!\(c\)) = x^2",DisplayFunction -> Identity],DisplayFunction -> $DisplayFunction],{c, -2, 2, 0.5}];

Looks familiar, but how long will predictability continue?  Chaos would say not long.  So, answer these familiar questions, before things change.

1) Desribe the changes taking place in the graph.
2) In terms of the general form, what value is being varied.
3) Does this fit that pattern that you would expect?
4) How would you determine this change if you were only given 
          an equation of a parabola?

Parabolas - Part 3 (The final chapter)

At this time one would expect little or no surprise at the next set of graphs.  However, expect the unexpected.  Without further delay, let's look at another set of parabolas.

Do[Show[Plot[((x^2)/c),{x, -10, 10},PlotStyle 
                    -> {RGBColor[0, 0, 1], Thickness[0.005]},PlotRange -> {{-10, 10}, {-10, 10}},AspectRatio -> Automatic,PlotLabel->"\!\(c\) y = x^2",DisplayFunction -> Identity],DisplayFunction -> $DisplayFunction],{c, 1, 6, 1}];

Seems to fit the expected pattern quite well.  Unlike circles, however (where a negative radius doesn't seem to make sense), we can substitute negative values.  This animation shows the result.

Do[Show[Plot[((x^2)/c),{x, -10, 10},PlotStyle 
                    -> {RGBColor[0, 0, 1], Thickness[0.005]},PlotRange -> {{-10, 10}, {-10, 10}},AspectRatio -> Automatic,PlotLabel->"\!\(c\) y = x^2",DisplayFunction -> Identity],DisplayFunction -> $DisplayFunction],{c, 5, -5, -2}];

Now, the questions would seem to be a bit more interesting:
1) What does the parabola appear to be doing?
2) What value is changing in the general form, causing this to 
          take place?
3) What predictions could you make about the graph, if given
          only the equation?
 4) How do your answers from these three parts fit together to 
           enable you to make a prediction about the size and 
           location of the graph of a parabola, given only its
           equation? (This is your idea of how to graph
           parabolas)
 5) What would be the difference if we were graphing the second 
           form of the general equation a(x-h)=(y-k) \(TraditionalForm\`\^2\), and             would the changes affect it in the same manner?  6) What are the similarities and differences between predicting             the graphs of circles and parabolas?

Ellipses

Just as an ellipse appears to be a circle that is a bit warped, the general form of the equations are similar in the same manner.  The general form for an ellipse is:
                  (x-h) \(TraditionalForm\`\^2\)/a \(TraditionalForm\`\^2\) + (y-k) \(TraditionalForm\`\^2\)/b \(TraditionalForm\`\^2\) = 1 At first glance it would seem that there would be 4 parts to the ellipse section, however, changing a and b would have similar effects.  Consequently, we will only work with one of them.

Ellipses - Part 1

Considering the last statement that was made in the introduction, let's start the study of ellipses with this set of animated graphs.

Do[Show[ParametricPlot[{2 Sin[t], c Cos[t]},{t, 0, 2 π},PlotStyle 
                    -> {RGBColor[0, 0, 1], Thickness[0.005]},PlotRange -> {{-6, 6}, {-6, 6}},AspectRatio -> Automatic,PlotLabel->"(x^2)/4 + (y^2)/\!\(c\^2\)= 1",DisplayFunction -> Identity],DisplayFunction -> $DisplayFunction],{c, 3, 6, 1}];

The hint above should have tipped you off that we are dealing with the combination of a and b, but one of the necessary question will be which value are we dealing with.

1) Describe the action in this sequence.
2) In terms of the general formula for an ellipse
          (x-h) \(TraditionalForm\`\^2\)/a \(TraditionalForm\`\^2\) + (y-k) \(TraditionalForm\`\^2\)/b \(TraditionalForm\`\^2\) = 1, which value is being           changed?
 3) What would be the effect if we were to change the other value           (either a or b) instead of this one? 
 4) What would be the rule for determining this change given only           the equation?

Ellipses - Part 2

While the major difference of ellipses has been exposed, we do still have some old friends to check up on.  The idea here is that we have learned that appearances can be deceiving.  Consequently, the next two sections will determine the level of consistency throughout our study of conics.

Do[Show[ParametricPlot[{2 Sin[t] + c, 3 Cos[t]},{t, 0, 2 π},PlotStyle 
                    -> {RGBColor[0, 0, 1], Thickness[0.005]},PlotRange -> {{-5, 5}, {-5, 5}},AspectRatio -> Automatic,PlotLabel->"(x - \!\(c\))^2/4 + (y^2)/9= 1",DisplayFunction -> Identity],DisplayFunction -> $DisplayFunction],{c, -2, 2, 0.5}];

Let's change some of the questions to see if we can get a better idea of what's going on.

1) What is the part of the general form that is being changed?
2) Would changing the values of a and b cause any changes in the 
          movement of this animation?  Why or why not?
3) What prediction would you make for the next section?

Ellipses - Part 3

After making your prediction, run this animation to see if it is a correct prediction.

Do[Show[ParametricPlot[{2 Sin[t], 3 Cos[t] + c},{t, 0, 2 π},PlotStyle 
                    -> {RGBColor[0, 0, 1], Thickness[0.005]},PlotRange -> {{-5, 5}, {-5, 5}},AspectRatio -> Automatic,PlotLabel->"(x)^2/4 + (y - \!\(c\))^2/9= 1",DisplayFunction -> Identity],DisplayFunction -> $DisplayFunction],{c, -2, 2, 0.5}];

I hope that your hypothesis was proven, and your prediction was correct.  This would verify that there is a degree of predictability in the changes that can be made in the values from the standard forms.  Now, a few final questions involving ellipses:

1) Was your prediction correct, and what pattern does it follow?
2) In putting the three parts together, what would be the rules
          to follow in predicting the size and location of the
          graph of an ellipses from its equation?

Summary Questions

To tie the information from circles, parabolas, and ellipses together, answer these questions.

1) The changes that were made could be separated into two 
          groups.  How would you classify those groups?
2) Which of those groups is consistent for the three types
          of conics that were studied?
3) What are those similarities?
4) Compare and contrast the other group of changes.
     a) How are they similar for all three conics?
     b) How do they differ for each conic section?