Assignment 10

Parametric Equations

by Jeff Hall

Problem 1

Graph the following equation:

Graphing the equation gives us this circle of radius 1 centered on the origin:

Problem 2:

Investigate the equation for various a and b constants in front of the t variables.

Let's explore other variations of this equation.

What would happen if we multiplied the first t by 2?

It creates a parabola. Now let's multiply the second t by 2:

We go back to creating a circle. No matter what constant is in front of

the t, as long as both parts have the same constant, a circle will be created.

Now let's try different constants, say a 3 in front of the first t

and a 2 in front of the second:

One might have expected a parabola again, but as you can see

we got someting quite different.
Problem 3

Investigate the following equation for various a and b:

Since we already know what the graph looks like when a and b equal 1,

lets try a and b equal to 2:

As might be expected, we have a circle of radius 2 centered around the origin.

Will we have a parabola again if a=2 and b=1?

No, this time we get an ellipse.

How about when a=1 and b=.5?

We can now see a clear pattern. The a variable determines the x-axis radius

while the b variable determines the y-axis radius.
Problem 4

Graph the following equation:

This creates the following graph.

Notice that the circle is not complete on the left side.

What can we do to fix it? Let's try extending the range of t

from -2 pi to +2 pi. Here is the graph now:

The gap grew a little bit smaller. Let's try t between -10 pi and +10 pi:

We finally have a closed circle.

Why was the circle not closed before? Your guess is as good as mine.

Now let's manipulate the equation to find out how it works.

First, let's try changing the constant in front of the first t variable, like so:

This produces the following:

The placement of 2 in front of the t made the blue shape cross the x-axis at -2

instead of at -1 as before. But the right side still crosses at x=1, so what should

we change to influence the right side of the figure? Let's try changing the first 1 to a 2.

It worked! By changing these two constants, we have influenced

both horizontal sides of the figure. But notice that the top and bottom parts

remained the same. The figure still crosses the y-axis at 1 and -1.

How would we change that? Let's try changing the constant in front of the t

in the y vector portion of the equation:

It worked! This time, since there is only one constant to change in the y vector,

the change from 2 to 3 changed the vertical diameter of the figure from 1 to 1.5.

Problem 5

Graph several sets of curves for:

Let me start by setting a and b equal to 1 and changing k. Here's the first graph:

Now let's change k from 1 to 2 and zoom out a little bit:

How about when k equals 5:

How about when k equals zero?

Finally, let's see what happens when k=-2:

It looks like changing the k value influences the slope of the line,

but the a and b values influence the starting point or common intercept.

Let's change a and b to see if this is correct. Let's make a=2 and b=2.

We should expect the line to cross (2,2). Let's make k=5 to make it easier to see:

As expected, the a and b variables determine where the line will cross, namely the point (a,b).
Problem 6

Graph the following:

I graphed the equation along t from -2pi to +2pi to get this:

What can we do to help us understand this equation better?

Let's try changing the constants from 1 to 0:

This changes the intercept point of the line from (1,-1) to (0,0).

Also notice that the red line crosses point (1, 2), meaning the slope is 2.

Changing the constants in front of the t's should change the slope:

As we would expect, the slope changed to 1/2 and crosses the point (2,1).
Problem 7

Write parametric equations of a line segment through (7,5) with slope of 3.

Graph the line segment using your equations.

As we saw in Problem 6, we know how to influence the slope and crossing point.

For this problem, let's try this equation:

The equation worked!
Problem 8

Investigate the following equation for different values of a and b:

What is the curve when a < b? a = b? a > b?

Let's try a=1 and b=2 first:

Now let's try a=b=1:

Finally, let's see what happens when a > b. Let a=2 and b=1:

As you can see, the a and b values influence the shape of what is a circle when a=b.

When b > a, the circle becomes an ellipsoid that is taller than it is wide.

When a > b, the circle becomes an ellipsoid that is wider than it is tall.

What would happen if the equation changed to this:

where h is any constant number?

Let's let h=1 and try the different values of a and b again.

When a=1 and b=2, we get this graph:

When a=1 and b=1, we get this graph:

And finally, when a=2 and b=1, we get this graph:

Notice that the angle of the ellipsoid is relatively similar for both b > a and a > b.

Now let's see what happens when h cycles between -3 and 3, and a=2 and b=1.

The last two parametric equations describe the locus of the vertex (x,y) of a triangle

with altitude h whose other two vertices are moved, one along the x-axis and

the other along the y-axis.

Problem 9

With t between 0 and 2pi, describe each of the following equations when a=b, a < b, and a > b.

Equation 1 is:

When a=b=1, we get:

When a < b, let a=1 and b=2 to get:

Finally, when a > b, let a=2 and b=2:

As we saw in Problem 8, this equation influences the shape of a circle turned into ellipsoids.

Equation 2

When a=b=1, we get this graph:

When a < b, let a=1 and b=2 to get this graph:

Finally, when a > b, let a=2 and b=1 to get this graph:

As you can see in these three graphs, the squaring of the cos and

sin functions in the vector produces lines connecting the axis intercepts.

Equation 3:

Now let's see what happens when you take the functions to the third power:

When a=b=1, we get:

When a < b, let a=1 and b=2 to get this:

When a > b, let a=2 and b=1 to get this:

These are interesting. The vector equations created diamond-like figures

when the equations were cubed. Notice that the a and b values influence

the height and width of the diamond, just like they did in the circle equations.

Taking the equations to the fourth- and fifth-power produce similar graphs

to the second- and third-power equations, except they become more curved.
Problem 10

Consider the following equation and graph from -20pi and +20pi:

Now consider a second equation:

Add this to the original graph to get this:

Problem 11

Consider the following equation:

When a/b = 1/2, we get this graph:

When a/b=1/4, we get this graph:

When a/b=2/3, we get this graph:

When a/b=12/13, we get this graph:

This class of parametric curves are called the Lissojous curves.

Let's compare them with this equation:

When a=1 and b=2:

When a=1 and b=4:

When a=2 and b=3:

When a=12 and b=13:

How do these compare?

Problem 12

A cycloid is the locus of a point that rolls along a line.

Write parametric equations for the cycloid and graph it.

Consider also a GSP construction of the cycloid.

Here are my equations: