Sketch the graphs of each of the following parametric equations. (a) x=3 cost, y=3 sint, 0 ≤t ≤2

Sketch the graphs of each of the following parametric...

Key Concepts

Spiral Curves

Spiral curves arise in parametric representations when one or both coordinates include a component that varies linearly with the parameter, typically multiplying a trigonometric function. This leads to curves that spiral outward or inward, exhibiting a continuous change in radius as the parameter increases.

Periodicity and Frequency

The periodicity of trigonometric functions plays a key role in the behavior of parametric curves. Modifications in the arguments of sine and cosine functions, such as scaling by different factors, affect the period and frequency, which in turn influence the number of cycles, loops, and self-intersections present in the graph.

Circular and Elliptical Graphs

When parametric equations involve cosine and sine functions with equal amplitudes, the resulting curve is a circle. If the amplitudes differ between the x and y coordinates, the curve is an ellipse. Recognizing these patterns allows for immediate identification of the basic geometric nature of the curve.

Parametric Equations

Parametric equations express the coordinates of the points on a curve as functions of a parameter, typically denoted by t. This approach allows one to describe curves that may be difficult or impossible to represent using a single Cartesian equation, providing flexibility in capturing a wide variety of geometric shapes and motions.

Conversion to Cartesian Coordinates

Converting parametric equations to Cartesian form involves eliminating the parameter to obtain a direct relationship between x and y. This often simplifies the analysis of the curve’s shape and properties by using familiar Cartesian forms such as circles, ellipses, or other conic sections.

Transcript

00:01 Okay, what we want to do is we want to go ahead and just kind of sketch a graph of these six parametric equations over these domains.

00:14 We're actually going to be using a graphing calculator.

00:18 So if you are using a graphing calculator, don't forget, someplace, they should have what is called a mode setting.

00:27 And under the mode setting, there should be a mode button.

00:30 Under the mode setting, you should see function, parametric, polar, sequential, something like that.

00:39 Make sure you under parametric mode in order for us to be able to graph these.

00:45 Okay.

00:46 I'm actually going to be using an online calculator called desmos, and i'll show you how to use that in just a minute.

00:55 Now, initially what we want to do is we're going to graph each of these separately and kind of talk about what they should look like.

01:06 Before we go into our graphing calculator, though, on some of these easier ones, you need to start recognizing.

01:13 So if you have x equal to a cosine of t and y equal to a sine of t, so the numbers in front of cosine and sine are both the same, then this will actually represent a circle.

01:25 With the radius equal to whatever that number is in front of the cosine and sign.

01:33 Now, if our numbers in front of cosine and sign are different, then this will actually be an ellipse.

01:43 And the major axis is going to be equal to the horizontal.

01:54 So that major axis is going to be the horizontal, and we have it three units long.

02:01 Well, actually, six units long, but three on either side of zero, zero.

02:07 And then, of course, this will be a one.

02:09 These other ones are going to be harder to recognize.

02:13 Now, if we do x squared plus y squared here, we get x squared plus y squared is equal to t squared.

02:21 So this is somewhat going to be a circle, but it's actually going to be a spiral because every time i increase t, then that radius of that circle increases.

02:31 So it's going to be kind of a spiral effect.

02:34 These others are kind of being harder to recognize.

02:37 So let's go ahead and go to our graphing calculator.

02:40 I'm going to switch to desmos.

02:41 Now desmos does not have a mode setting per se.

02:47 So we actually have to kind of graph what are called our postmos.

02:52 Parametric points over that region.

02:55 And so we actually put in points here.

02:58 So this would be our x value.

03:00 This would be our y value.

03:01 So don't forget, our first x value was 3 cosine of t.

03:10 And, whoops, i want to put a parentheses around here.