In this lesson we will use desmos.com
to explore the Archimedean Spiral,
given by the polar equation r equals n theta.
To begin, let's go to desmos.com
and click graphing calculator.
In cell one, enter r equals n theta.
To enter theta from the keyboard type in T-H-E-T-A
or using the Desmos keypad, click ABC,
and theta is on the far right.
Notice how we also have the option to add a slider for n.
Let's go ahead and do that now by clicking the n,
and notice how we have a graph of a spiral.
The interval for theta is from zero to 12 pi.
So keep in mind for this exploration,
theta is always greater than or equal to zero.
Before we use a slider for n though,
let's change the grid from rectangular to polar,
by clicking on the wrench in the upper right hand corner,
under grid, click the polar graph.
We also have the option for radians or degrees.
Let's stay in Radian mode,
and now we click outside this menu to close the menu,
if we want, we can zoom in or zoom out
using the plus or minus buttons on the far right.
Minus zooms out, plus zooms in.
Let's go ahead and use this window here for right now,
before we explore how n affects the graph.
Let's trace out this graph from let's say zero to four pi.
To do this, let's click in cell one,
click and hold the red circle on the left.
Let's change this graph to dashed,
and change the color to orange.
Click outside this menu.
Now we want to duplicate this equation by clicking the gear,
and then click the duplicate option,
and now we will go to cell two.
Let's change the style of this graph
by clicking and holding the orange circle on the left.
Let's change the style back to solid,
and the color to purple.
Click outside this menu.
And now we are going to restrict the domain
for the variable theta from zero to a,
then create a slider for a.
To do this we have to use the squiggly brackets,
which using the Desmos keypad is here toward the bottom.
So click the squiggly brackets,
and now we are going into enter zero,
less than or equal to theta.
Less than or equal to a.
Notice now we have an option to add a slider for a.
Let's go ahead and do this by clicking the a,
and now let's change the interval for a
from zero to four pi.
Click the negative 10.
Enter zero tab four pi.
For pi we can enter a pi from the keyboard,
or from the number keypad.
Pi is here toward the bottom tab.
Let's have the step b, pi divided by 12.
Click outside cell three.
Let's change the value of a to zero
by dragging the slider to the left.
And now we can drag the theta to the right,
and see the spiral traced out from zero
to four pi radians.
Notice how the spiral is traced out
in the counterclockwise direction.
Notice how we have two full, complete revolutions
of the spiral,
because the interval is from zero to four pi.
If we want the entire spiral,
we'd have to change the upper bound of a,
from four pi to 12 pi.
Let's go ahead and do that.
Click on the floor pi.
Enter 12 pi.
Click outside cell three.
Close the Desmos keypad,
and it continues sliding a to the right.
We can always zoom out or zoom in if we wish.
Zoom out enough, we can drag all the way to 12 pi,
to get the spiral from zero to 12 pi radians.
Let's go ahead and zoom in,
and now let's see how the value of n affects the spiral.
Keep in mind we're only using the angle theta from zero
to 12 pi radians, meaning theta is non-negative.
So let's change n to zero.
The graph disappears and notice as n increases,
the spiral enlarges,
or we can say the points get further and further
from the pole.
Notice how the spiral is in the counterclockwise direction.
So when both theta is positive and n is positive,
the spiral is counterclockwise.
And now let's change n back to zero.
Now let's move n in the negative direction.
Notice when n is negative,
the spiral is now in the clockwise direction,
and we can say as the absolute value of n increases
the spiral enlarges, or the points get further and further
from the pole.
So again, when theta is positive and n is negative,
the spiral is clockwise.
So the sign of n and theta determine the direction
of the spiral, and they also determine
how far the points are from the pole
or how large, or how small the spiral is.
Let's summarize our findings.
The polar equation r equals n theta
is the Archimedean spiral.
As the absolute value of theta increases
the distance from the point to the pole increases.
As the absolute value of n increases
the distance from the point of the pole also increases.
If n times theta is positive or greater than zero,
the spiral of is counterclockwise.
And if n times theta is less than zero or negative,
the spiral is counterclockwise.
I hope you found this helpful.