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For more information, please see full course syllabus of Pre Calculus
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Lecture Comments (2)

1 answer

Last reply by: Professor Selhorst-Jones
Sun Apr 20, 2014 8:47 PM

Post by Taylor Wright on April 19, 2014

Thank you for this amazing lecture series over PreCalc!!!  Do you know if there are any plans to incorporate any Engineering specific lectures in the near future such as statics, dynamics, fluids, etc.?

Thank you!

Parametric & Polar Graphs

  • If you want to graph either a parametric or polar graph, the first step is to change the graph type your calculator is currently using. This option is often in the general `settings' , but it will vary from calculator to calculator. Once you find where your calculator has this option, change it to whatever type you want to work with.
  • You still have to set up your viewing window when working with parametric and polar graphs. You need to choose the appropriate xmin, xmax, ymin, and ymax to view your graph. (Or move it to an appropriate place using zoom.)
  • Setting up a parametric graph is very similar to setting up a "normal" function graph. Instead of setting up a single function though, you set up both a horizontal function x(t) and a vertical function y(t). The only major difference is that you have to tell the graphing calculator what interval the parameter should use. It will often default to t:[0, 2π] or t:[−10, 10], but for some graphs that won't be enough.
  • Graphing in polar is very similar. We enter some function r(θ) in terms of our independent variable θ. Just like parametric, we have to pay attention to the interval our θ is given.
  • When graphing both parametric and polar functions, you might see that the calculator doesn't produce a very smooth graph. That's because of the tstepstep: the step-size between points it uses for graphing. If you use a smaller value, the graph will smooth out.

Parametric & Polar Graphs

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Change Graph Type 0:08
    • Located in General 'Settings'
  • Graphing in Parametric 1:06
    • Set Up Both Horizontal Function and Vertical Function
    • For Example
  • Graphing in Polar 4:00
    • For Example

Transcription: Parametric & Polar Graphs

Hi--welcome back to Educator.com.0000

Today, we are going to have the last lesson in our graphing calculator appendix, Parametric and Polar Graphs.0002

If you want to graph either a parametric or a polar graph, the first step is to change the graph type your calculator is currently using.0007

This option is often in the general settings on your calculator, but it will vary from calculator to calculator.0015

Once you find where your calculator has this option, change it to whatever type you want to work with.0020

We will look at working with parametric first; then we will look at polar.0024

And notice that you still have to set up your viewing window when you are working with parametric and polar graphs.0028

You need to choose the appropriate xmin, xmax, ymin, and ymax to view your graph.0033

You need to have that horizontal and vertical location, so that you have a viewing window to look through.0039

Or if you don't want to use xmin, xmax, ymin, and ymax, setting it by hand, you can also just move it to the appropriate place using zoom.0043

Start your graph, and then use the zoom function to back it out, and then put it into whatever place you want to look at.0050

However you want to do it, you just still have to deal with the viewing window, just like you did with a normal function graph.0056

All right, setting up a parametric graph is very similar to setting up a normal function graph.0062

Instead of setting up a single function, though, we will be setting up two functions, a horizontal function x(t) and a vertical function y(t).0067

So, you set each of those up, just like you used to set up one function,0075

but now you are setting up how your horizontal part moves and how your vertical part moves.0079

The only major difference is that we have to tell the graphing calculator what interval the parameter should use.0083

It will often default to t going from 0 to 2π; and on some calculators, it might default to t going from -10 to 10.0089

But for some graphs, that won't be enough; you will have to think about the specific one you are using.0096

If it looks like it is cut off all of a sudden, you might want to just expand your interval to check and see if that fixes things,0100

if that changes the graph you are looking at.0106

Or if you think about it a lot beforehand, you might be able to say, "Oh, I see what kind of interval will fit my viewing window well."0108

So, that is something to think about: that the interval that you choose for your parameter will affect it,0115

and the default one isn't necessarily always going to work.0120

For example, if we graphed x(t) = t + 1 and y(t) = (t - 1)2 - 50124

with the default interval of t going from 0 to 2π, we would obtain this graph here.0131

Oh, it is missing a big chunk of the graph, because of that default interval!0136

We can see that it should also have a portion going this way, but it is just completely missing that,0140

because our default interval stops--it doesn't even go under 0.0145

So, because of that, we have just cut off this large chunk.0150

So, that default interval can really cause some problems.0152

If we want to be able to see the whole thing, we need to change our interval.0155

To fix the missing portion, we increase the size of our intervals to obtain this new graph.0159

So, if we switch to t going from -10 to 10, it would now go actually past the edge of our viewing window,0163

and we have completely filled what our viewing window can see of this graph.0168

So, that is much better; but I want you to notice that the graph here isn't quite smooth.0171

We sort of see these jagged corner edges in parts.0176

That is because we are using a large tstep; tstep is a specific value that says the step size between points it uses for graphing.0180

The way a graphing calculator graphs things is basically the way we graph things.0188

It evaluates multiple points, and then it draws a curve between those points.0192

But because we are evaluating multiple points, and then it is drawing a curve,0197

if there is a lot of space between these steps in those points, it will end up getting these jagged edges,0202

where it doesn't quite know how to make the appropriate curve.0207

So, if you want to be able to get a nice, smooth curve, you need to use a smaller tstep value.0210

You lower the size of the step between the points, and it will make a nice, smoother curve, just like when we are graphing ourselves.0215

If we want to get a better sense for how the graph is going to work out, we want to have less space between the points that we put down.0220

So, we do the same thing for our graph: we use a smaller value, like tstep = 0.05.0226

And now, our graph looks nice and smooth; the graph smoothes out,0231

because there are now enough points on the graphing calculator for it to be able to make a smooth curve for us to see.0234

Graphing in polar is very similar: we enter some function r(θ) in terms of our new independent variable, θ.0240

Just like parametric, we have to pay attention to the interval our θ is given.0247

The interval will normally, standard-ly go to t going from 0 to 2π.0251

But often, that is not going to be enough for some of the functions that we will be working with.0255

And occasionally, we will want to deal with the θstep value, as well, if we want to smooth it out.0259

It might end up being jagged for certain initial θstep values, so we might want to make it smaller to smooth things out.0263

For example, if we had this one right here, graphing r(θ) = θ with a default interval of t going from 0 to 2π, we would obtain this graph here.0269

But this graph is completely missing lots of information.0277

We might be able to realize that it actually could keep going that way if our t was allowed to go to a larger thing.0279

Oh, t should actually be θ in this case, for the specific thing we are using here, since I set r(θ) here.0285

However, with some graphing calculators, it will actually end up using t for polar, as well; it depends on the specific calculator you have.0292

Most calculators, though, will use θ for this.0298

So, in this case, we have our default interval of θ going from 0 to 2π, and we realize that there is stuff missing here.0301

If we were able to go to lower θ values, we would be able to get different stuff.0308

If we had higher θ values, we would have other places to graph.0312

So, we have to expand our interval; we expand our interval...and once again, that should be a θ...0314

and we get this graph here, and so we have a much better idea of what the thing looks like.0322

We see, though, that it doesn't look quite as smooth as we want, so the issue there is our θstep.0328

So, we can go back and choose a small θstep; we can smooth it out by putting a θstep of 0.05; and now it is smoothed out.0333

One thing to notice, though: our θ of going from -10 to 10...we see in this viewing window0342

that we are actually still not completely using everything that can go into this viewing window.0346

If we were to increase our interval to a lower starting value and a higher starting value, we would actually end up continuing this out.0351

We are only seeing a portion of the graph; this will continue to spiral out forever and ever.0360

And so, we might want to increase our θ even larger, so that we can completely fill the viewing window.0365

It depends on the specific situation, but it is definitely something to think about.0370

So, when you are dealing with parametric and polar graphs, it is very, very similar to graphing a normal function.0372

But now you have to pay attention to what the interval is that you are graphing.0378

With a normal function, it graphs the entire x interval; that is pretty easy.0381

So, you know that you have all of the things that you could be interested in, since it is all of the x-values.0385

But with t and θ, it is something where it is not quite directly what we are looking at in the viewing window.0389

So, we have to set this arbitrary interval; with t, you really have to think about what will be useful stuff here,0397

and the same with θ--what will be useful stuff here?0403

With lots of polar things, it will end up repeating, so 0 to 2π will be enough.0405

But sometimes, you need larger things before it ends up repeating.0409

And sometimes, it won't repeat at all, at which point you want to just keep expanding your interval until you have filled out your viewing window.0412

Think about the interval; and occasionally, if it is kind of rough around the edges, just lower your θstep or your tstep value,0417

so that it smoothes out and you get a nice, smooth curve.0423

All right, that finishes it for graphing calculators; we will see you at Educator.com later--goodbye!0425