Welcome to a video
that will show how to graph parametric equations
on a TI83 or 84 graphing calculator.
The goal of the video is to graph and determine points
of a plane curve expressed as parametric equations.
The first example would be X equals four minus T,
Y equals 2T.
Notice there's no interval expressed for T,
so we assume T would be all real numbers.
First thing we need to do is put our calculator
in parametric mode.
If we press the mode key,
we need to go to row four
and highlight PAR for parametric.
So we can press down,
and then right once over to parametric.
Press enter, and now it's in parametric mode.
So now if we press Y equals,
notice we have X1 of T and Y1 of T.
So our first example was X equals four minus T.
So now if I press the variable key,
T will show up.
Press enter,
and then Y was equal to 2T.
Enter.
The next thing we have to do is set up the window.
Notice the first option for the window
is T minimum and T maximum.
For this example, T can be any real number,
but we still have to define a T minimum and a T maximum.
So let's go ahead and select
from negative 10 to positive 10.
Next, we need to select Tstep,
which you can think of as a change in T each time.
Let's go ahead and change this to .1.
If our Tstep is too large, the graph will not look smooth.
And then the Xmax, Xmin, Ymax, Ymin
is the same as it normally is
setting up our window to plot the order pairs XY.
Now let's go ahead and press graph.
We can see our graph is a line.
Now if we press trace,
notice how the T starts at negative 10,
and if we press the right arrow,
you can see T increases by .1 each time.
Now these points are off the view of the screen.
Once T increases to negative five,
you can see this is the point 9 negative 10.
So if we had to indicate the direction,
this line would be traced, as T increases,
we can see it would be in this direction.
So we can find points to plot on our paper
using the trace feature,
or we can use the table features of the graphing calculator.
Let's go ahead and set up our table.
We'll press second window.
Let's go ahead and change our table start to zero,
and we'll have the change in the table by ones.
Let's go ahead and leave the independent variable
on automatic.
And now we'll press second graph.
And here it shows the values of T
and the corresponding values for X and Y for our graph.
Now let's take a look at X equals the square root of T
and Y equals five minus T.
Notice here, T has to be greater or equal to zero,
because X is equal to the square root of T.
Notice we have our old equations here,
so we can just press clear enter.
And now we have X equals the square root of T.
So second X square brings up the square root.
T, close parenthesis,
and then Y was equal to T minus five.
Now we get to go back and set up our window.
We can't leave our T minimum at negative 10
because X would be undefined.
Remember T was on the interval from zero to infinity,
so we'll change T minimum to zero,
press enter,
and we'll go and leave everything else the same.
Let's press graph.
Now, notice that our graph stops right here,
and that's because we had our T maximum set at 10.
So what we want to do is see more of this graph.
We would have to increase the T maximum.
Let's go ahead and press trace.
So when T is zero, X is zero, and Y is negative five.
We press the right arrow.
We can find additional points on this graph.
Or if we go to the table by pressing second graph,
we can also select the points that we want to plot
from this screen.
You can see from this graph,
it looks like it's half of a parabola that opens up.
Okay, let's go and take a look at
a couple more interesting graphs now.
For these next three examples,
we're going to consider T on the close interval
from zero to two pi.
And for number three, we have X equals five sine T,
and Y equals two cosine T.
So we press Y equals, let's clear these.
And again, X is five sine T,
and Y equals two cosine T.
We do need to go back and adjust the window,
so press window.
Remember, an interval for T was from zero to two pi.
So we'll leave Tmin at zero, and Tmax is going to be two pi.
So we can press the two,
and then second exponent key brings up pi.
I'm also going to change the T step.
I'll change just the pi divided by 12.
Before we graph this, let's make sure we're in radian mode.
And it looks like we are in radian mode.
Let's go ahead and press graph.
And you can see here we have an ellipse.
The major axis would be 10 units,
and the minor axis would be four units.
If we press trace, notice that when T is zero,
it starts right here at the .02.
And as T increases, it's going to be more difficult to
plot these points accurately.
But you can see this ellipse is traced clockwise
as T increases from zero to two pi.
Or of course we could press second graph
to access the table feature as well.
Let's go and take a look at two more.
Our next example is X equals six, cosine cube T,
and Y equals six sine cube T.
So I'll go ahead and clear these.
So we have six, and then cosine cube T,
and Y was six sine cube T.
The interval for T is the same,
so let's go ahead and press graph.
And you can see we have this asteroid shape plane curve.
If we press trace
and then the right arrow,
we can see that this plane curve is traced out
counterclockwise
as T increases from zero to two pi.
And the last one, we have X equals four cosine T,
and Y equals two sine four T.
So X equals four cosine T and Y equals two sine four T.
Let's go ahead and press graph.
And you can see we get a very interesting looking graph
from these parametric equations.
And if we press trace one more time,
we can see how this curve is traced
as T increases from zero to two pi.
Okay, that's going to do it for this video.
I hope you found it helpful.
Thank you for watching.