### Mike Cotton

A parametric equation in the plane is a pair of functions   where the two continuous functions define an ordered pair (x,y). The two equations are usually called the parametric equations of a curve. The extent of the curve will depend on the range of t.

Part I

Let's take a look at the parametric equation , with the interval . The graph of this equation can be seen in Figure 1 below.

This graph was created by Graphing Calculator 3.2.

Figure 1

Keep in mind that the curve above is over the interval . That is, if the interval was then only the bottom half of the circle would be the curve. See Figure 2 below.

This graph was created by Graphing Calculator 3.2.

Figure 2

If the interval is , then the curve would start at (1,0) and create 2½ revolutions and end at (-1,0). The list below shows the values of x and y that correspond to multiples of  π/4 for the parameter t.

 t x = cos t y = sin t 0 1 0 π/4 1/√2 1/√2 π/2 0 1 3π/4 -1/√2 1/√2 π -1 0 5π/4 -1/√2 -1/√2 3π/2 0 -1 7π/4 1/√2 -1/√2 2π 1 0 9π/4 1/√2 1/√2 5π/2 0 1 11π/4 -1/√2 1/√2 3π -1 0 13π/4 -1/√2 -1/√2 7π/2 0 -1 15π/4 1/√2 -1/√2 4π 1 0 17π/4 1/√2 1/√2 9π/4 0 1 19π/4 -1/√2 1/√2 5π -1 0

Part II

Let's take a look at the parametric equation , with the interval . Let b = 1, and let a vary.

This graph was created by Graphing Calculator 3.2.
Figure 3

Since a doesn't equal b the result is an ellipse. The list below shows the values of x and y that correspond to multiples of  π/4 for the parameter t.

 t x = cos t x = 2cos t x = 3cos t x = -4cos t y = sin t 0 1 2 3 -4 0 π/4 1/√2 2/√2 3/√2 -4/√2 1/√2 π/2 0 0 0 0 1 3π/4 -1/√2 -2/√2 -3/√2 4/√2 1/√2 π -1 -2 -3 4 0 5π/4 -1/√2 -2/√2 -3/√2 4/√2 -1/√2 3π/2 0 0 0 0 -1 7π/4 1/√2 2/√2 3/√2 -4/√2 -1/√2 2π 1 1 3 -4 0

What is interesting to note is that for a = 2 & 3 the curve starts at (1,0) and goes counter clock-wise, where the curve for a = -4 starts at (-4,0) and proceeds clock-wise. Also, the equation of an ellipse is  .  Multiplying the equation
by a2b2 we get the equation .  If a = b then we get the equation x2 +y2 = a2 which is the equation for the circle.

Part III

Let's take another look at the parametric equation , with the interval . This time let a = 1, and let b vary.

This graph was created by Graphing Calculator 3.2.
Figure 4

Again, since a doesn't equal b the result is an ellipse. The list below shows the values of x and y that correspond to multiples of  π/4 for the parameter t.

 t x = cos t y = sin t y = 2sin t y = -3sin t y = 4sin t 0 1 0 0 0 0 π/4 1/√2 1/√2 2/√2 -3/√2 4/√2 π/2 0 1 2 -3 4 3π/4 -1/√2 1/√2 2/√2 -3/√2 4/√2 π -1 0 0 0 0 5π/4 -1/√2 -1/√2 -2/√2 3/√2 -4/√2 3π/2 0 -1 -2 3 -4 7π/4 1/√2 -1/√2 -2/√2 3/√2 -4/√2 2π 1 0 0 0 0

What is interesting to note is that for b = 2 & 4 the curve starts at (1,0) and goes counter clock-wise, where the curve for b = -3 also starts at (1,0) but proceeds clock-wise.