For more information, please see full course syllabus of Math Analysis

For more information, please see full course syllabus of Math Analysis

### 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
`tstep`/θ`step`: 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

### Math Analysis Online

### 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} - 5*0124

*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 window*0342

*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

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!