7.1 Parametric Equations
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Instructor: Now on example 1 here, we're given two equations, one to represent the x values, one to represent the y values, and a range for t values. So what we're going to do is begin with a chart. We have our t values, our x values. And I'll go and call it x of t and y of t.
Now we're going to evaluate from negative 3 to 2. So I'll just go and take these integer values there going to 2. Now we want to plug the t value into each of those individual equations. So if t is negative 3, then the x value is negative 4 because it's t minus 1.
I'm just going to go do this for the rest of these. Negative 2 would be negative 3, negative 1 would be negative 2, 0 would be negative 1, 1 would be 0, and 2 would be a value of 1. Do the same thing to get the y values here.
If I plug in negative 3, OK, let's see, that would be negative 6 and negative 2. Plugging in a negative 2, my t value there, negative 2 would give me 0. Negative 1 value there would give me 2. Negative 1 would give a positive-- no, 0 would give me a 4.
We're plugging in our t values. Value of t is 1, would give us a 6. Value of t equal to 2 would give us an 8.
So the points that I'm going to plot are these-- negative 4, negative 2, 4, negative 2, negative 3, negative 0-- or positive 0-- neither, 0, negative 2, 2, negative 1, 4, and 1, 8.
So our function appears to be linear here. I'm going to go ahead and put arrows on this indicating that as t increases we begin from one end and go to the other. And that is our parametric curve.
Now let's do the same thing for this next one, x of t equals t squared minus 3, y of t equals 2t plus 1. And our values of t are ranging from negative 2 to 3.
So negative 2, negative 1, 0, 1, 2, 3. And our x values are going to be 1. Plugging in a value of negative 1, we should get negative 2. Plugging in 0, we should get negative 3. And yes, that will work.
Plugging in a value of 1, we get negative 2. Plugging in a value of 2, we get 1. Plugging in 3, we get 6.
All right, now this next one is linear. So plugging in a 0 would actually get me 1. And the slope is 2, so this should be 3, 5, 7, and that'd be negative 1, negative 3. Increasing by 2 going up, decreasing by 2 going down.
Now to plot these points, I have 1, negative 3. I have negative 2, negative 1, negative 3, 1, negative 2, 3, 1, 5, and 6, 7.
1, 2, 3, 4, 5, 6. So 5, there we are. So beginning on this end, it appears that we are going to have a smooth graph here, ending there. And it is going in this direction.
So as we increase our values of t, we start at this one endpoint down here, 1, negative 3, and we increase up to 6, 7. And there's our curve. And there's the graph that those equations represent.
All right, example c, x of t equals 4 cosine t, y of t equals 4 sine t. And t ranges from 0 to 2 pi. So again, go ahead and take our t values, our x of t and our y of t.
All right, I'm going to begin at 0, going to go up to 2 pi. I'm going to go in pi over 2 increments. You can go in whatever increments you'd like. That's what I'm going to go with.
All right, so if we evaluate, cosine of 0 is 1, so that would give us 4. Cosine of pi over 2 is 0. Cosine of pi is negative 1, so that would make it a negative 4 there. 3 pi over 2, cosine is 0. Cosine of 2 pi is 1, so that would be 4.
Doing the same thing for sine-- sine of 0 is 0, so we get a value there. Sine of pi over 2 is 1, so that would be 0, 4. Sine of pi is 0. Sine of 3 pi over 2 is negative 1, so that would be a negative 4. And back at 2 pi, same thing as 0, so we have 0 there.
So plotting these points, I have 4, 0, 0, 4, negative 4, 0, and 0, negative 4. And then we come back to that point. Now I'm going to attempt to draw this smoothly. I'm not going to give it justice, I know.
So now if I increased or decreased my increments there, I should see that this is actually a circle. This is a circle. And we are moving in this direction. We are going counterclockwise.
And that would be our curve. So our curve is a circle-- or in my graph, something more of a rounded circle.
[ENDS AT 8:05]