How to Use the MacBook Grapher to Graph Parametrized Lines and Surfaces in 2D and 3D

Parametric Curves and Surfaces with MacBook Grapher

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How to Parameterize Curves and Surfaces Using MacBook Grapher

This lesson demonstrates how to use the built-in MacBook Grapher software to create and visualize parametric curves, surfaces, circles, spirals, and three-dimensional objects.

The lesson begins with a standard parabola:

$y=x^2+1$

The equation is then parameterized using:

$x=t,\qquad y=t^2+1$

with parameter restrictions such as:

$0\le t\le10$

Two Dimensional Parametric Curves

Students are shown how to enter two-dimensional parametric equations directly into MacBook Grapher by selecting the parametric graph option and replacing the default expressions with custom equations.

The lesson demonstrates how restricting the parameter changes the visible portion of the graph and how frame limits can be adjusted to zoom in and out of the graphing window.

Example:

$x=t,\qquad y=t^2+1,\qquad 0\le t\le10$

Three Dimensional Curves

The lesson then extends the parameterization into three dimensions by introducing:

$z=0$

which creates a three-dimensional curve lying in the plane:

$z=0$

Students are shown how one parameter creates curves while multiple parameters generate surfaces.

Parameterizing Surfaces

The lesson introduces surfaces using two parameters:

$x=uv,\qquad y=u,\qquad z=uv^2$

with parameter intervals such as:

$0\le u\le100,\qquad 0\le v\le100$

This demonstrates how surfaces can be generated and visualized directly inside Grapher.

Parameterizing Circles

The lesson also demonstrates parameterizing a circle using trigonometric functions:

$x=\sin(u),\qquad y=\cos(u),\qquad 0\le u\le2\pi$

Students observe how changing the interval from:

$0\le u\le\pi$

to:

$0\le u\le2\pi$

changes the graph from a semicircle to a full circle.

The Helix Example

The lesson introduces the helix using:

$x=\sin(u),\qquad y=\cos(u),\qquad z=u$

which creates a spiral structure commonly referred to as a helix or double helix in scientific and medical contexts.

Using MacBook Grapher Professionally

A major point throughout the lesson is that MacBook Grapher is an extremely powerful visualization tool already built directly into macOS.

Jonathan explains that students can:

Create graphs quickly.
Model curves and surfaces.
Visualize parameterizations.
Take screenshots directly into Microsoft Word.
Use the software to verify homework and textbook problems.
Create professional looking STEM documents.

The lesson encourages students to combine:

MacBook Grapher
Microsoft Word Math Print
WolframAlpha

into a complete technical workflow for mathematics, physics, engineering, and scientific communication.

Final Message

The lesson concludes by encouraging students to use graphical software as a tool for understanding and verifying mathematics rather than relying only on memorization.

Students are encouraged to continue experimenting with curves, surfaces, spirals, parameterizations, and three-dimensional visualization techniques as they progress into higher mathematics and multivariable calculus.

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Transcript reference: :contentReference[oaicite:0]{index=0}

Original Transcript

All right, kids. I’m going to show you guys how to do parametric equations here in uh the grapher with your Mac Mackie bookie. So, I’ve got my uh a default document here. Let’s let’s just take let’s say y is equal to x^2 + 1. And we want to parameterize that. We would say x= t and then y is equal to t ^2 + 1. and then we’ll say t is in uh real numbers. Okay, so that parameterizes it. It’s a really basic parameterization. Uh there’s more complicated parameterizations and we’ll look at two and threedimensional situations here. So I’m going to go to my grapher here. This is this grapher is built into the uh MacBook here. I just got a whole bunch of stuff open up here and it’s just it’s just uh I was just doing a whole bunch of stuff and where’s the graph at? It’s causing some issues.

Okay, let’s open this up. When you’re doing a lot of stuff all at once. Okay, we’ll just hit reopen then and close. Okay, so hit reopen. That caused an issue. Don’t save. I was doing a bunch of stuff. Okay, so so I want to do par I want to parameterize y is equal to x^2. Okay, y equ= x^2 + 1. There’s your parabola on all real numbers. Now I want to parameterize it. So I am going to go to new equation and I am going to go here over here and I’m going to click for two dimensions. I’m going to click that. I’m going to highlight that. I’m going to highlight all that. I’m going to get rid of that and do it like that. Then I’m going to let x be equal to t and y be equal to t ^2 + 1. So say I want to parameterize it on the um quadrant 1 from 0 to 10. So I do t is equal to dot dot dot 0 dot dot dot to 10. Now you see that that half of it there. So let me uh let me go to let me recolor this here. And I also like to get rid of the axis. I I don’t like the the graphing paper basically. I don’t like that axis in there. It’s distracting. Not necessary. Uh where’ that go? Here. So here. So let me let me zoom in on this a little bit. And uh zoom out. Whatever. We can change the uh let’s change the the view uh the frame limits. Let’s change it to negative one to we’ll keep it at five. And then we’ll go from zero. We’ll go from negative one in the Y, negative one in the Y, and one in the Y like that. And that’s not high enough. So, let’s go back to the frame limits. We want to probably start, we’ll make this 10. Uh, sorry, 10 on the Y, not one on the Y, 10 on the Y. That’s what I wanted. So there there’s if I wanted to parameterize this equation just in the uh in quadrant one, that’s how I would do it. If I want to if I want to take it out into the other quadrants, I just add in the rest of it. You can just see how the numbers change. So now let’s say I want to do this in three dimensions. I want to do the same exact equation in three dimensions. Then we’re going to go from a line to a plane. In this case, it’ll be like a trough. So I will make we’ll make z equal to zero since there was no z. y is equal to t ^2 + 1 and then x is equal to t and I hit enter. I get nothing right because I got to make I got to set my parameters. So t is equal to 0 dot dot dot 100. So there’s a did I say a plane? That’s going to be a line. A plane a plane would have two variables. So like uh uv. So let’s do it this way. UV and then U. We’ll do UV and UV^ 2. UV and UV 2. And then I’ll change this to to U. And then V will be equal to 0 dot dot dot 100 as well. And so there’s your surface. Did I say plain surface? Same. Well, you know, a curved plane surface, whatever. So that is uh just a snippet of something. I don’t know what that is. I just made stuff up, you know. So let’s let’s make it a little bit bigger here. Stretch it out. Cubed. Bigger. Okay. So there’s there’s a parameterization of a surface. So if you’re parameterizing surfaces, there’s multiple variables. If you’re doing a line in 3D is what I meant to say. I’m just waking up. I got my words mixed up here. If you’re going to do a line, you don’t have to use T. You can just we could just stick with U. And I want to go back and parameterize a curve. Maybe I want to parameterize a circle. So I’ll do sin u and cosine u. And uh we’ll go from zero to to pi. See what we got here. I don’t have anything in there because well hold on. Zoom in. Zoom out. There it is. You see it? There’s my circle. All right. There’s my line right there from 0 to pi. I meant to say 2 pi gives me the full circle. So there’s the full circle. So now what if I put a v? Let’s put a v down here. Let’s put u down here. That lifts it up into a what is that called? The um uh what’s it called? The spiral. The the medical term

when they’re twisting around each other. That’s a great that’s a great question to use chat GBT for. It’s right on the tip of my head. What is the 3D line curving cir curving around curving around itself called same one for medical logo helix.

Bam. I remembered right before. See a double helix. A helix is what I was right in my head. It was right there and it came to me. So helix is um that now what if if we’re talking about surfaces then we’re going to be in integrating um we’re going to be integrating

another variable in there. So let’s add let’s add v [clears throat] in there and we’ll do uv and hit enter. So now you’ve got this surface that’s like that. You can model different surfaces that way. So, if you have an equation or something, you’re trying to model something, you can do this to model things. You know, I could draw a coffee cup in here and model it and create a system of equations. There’s all kinds of different ways to do equations and things in here. Th this is a very powerful software. It’s built into the MacBook. And using this, using this along with Microsoft Word is fantastic because you can do this and just take screenshots and then just go and add it to your document. So, here’s the here’s the equation. that we were doing and there’s whatever and you can add it to your document. Why you would ever want to use anything other than this is beyond me. It’s it’s um whatever. So anyways, that’s that’s how you use that’s how you use um the MacBook Grapher to parameterize curves and surfaces. It obviously I did very simple examples. I’m just showing you how to type it in so that when you’re doing 3D calculus that you can uh you can check your work in here and I’ll be doing a lot of that coming up. So stay tuned. Don’t forget to subscribe. Pick up one of my books if you can. I appreciate donations. You guys’ support is what keeps this going. Become a member to the YouTube channel if you can.