ASSIGNMENT 10

BY

SHADRECK S CHITSONGA

PARAMETRIC CURVES

Let us start by doing a quick review of the ordinary equations.

y = mx + c is the general form of a linear function. We know that when we plot this function in the Cartesian plane we get a straight line. This was explored in assignment one on graphs of linear functions. The same is true with the quadratic functions, which we explored in assignment 2.

Now that we are familiar with these functions and their graphs, we want to switch our attention to parametric equations.

But what are parametric equations and what makes them different from the other equations.

Let us look at the following functions,

(i) x = t - 3

(ii) y = 2t + 5 , we will plot these equations. t ranges from -7 to 10. Refer to figure 1.

Figure 1

Now let us compare that with the graph shown in figure 2

This is the graph of y = 2x +11

Figure 2

By plotting this pair of functions in which both x and y are linear functions of t , we end up with a straight line. Plotting the equation y = 2x + 11 seems to give us the same line. The only difference is that the first line is shorter while the second one seems to extend from negative infinity to positive infinity. Remember we have put some restrictions on the value of t, t can only vary from -7 to 10. If we were to change these values the length of the line would also change. But how are the two line related?

For the first line we have three unknown variables t, x and y. The two equations that we used to plot this line are called parametric equations. And t is the parameter. There are different types of parameters. A parameter could be an angle or a length.

There are two basic conditions that a parameter must satisfy:

1. Each point on the curve must be related to a unique value of the parameter.

2. Each value of the parameter must give the coordinates of only one point on the curve.

In general

A parametric curve in the plane is a pair of functions

x = f (t)

y = g (t)

It is possible to derive the Cartesian equation from the parametric equations. Let us just look at a simple example.

(i) x = t - 3

(ii) y = 2t + 5

We can express equation (i) in terms of t, therefore we see that t = x + 3,

Substituting that in (ii) we get y = 2 (x + 3) +5 = 2x +11

Now we can see why we get the same line apart from the different lengths.

We will now extend our discussion and look at the trigonometric functions.

1.Let us consider y = sin (t)

x = cos (t ) for 0 < t< 2pi

Figure 3

Plotting the parametric equations y = sin (t)

x = cost (t ) 0 t 2pi

we get a circle with radius one. Does that surprise you? Let us look at the equation of this circle, Is there any connection between the equation of the circle and our parametric equations?

and . Using the famous trigonometry identity we see that .

Just for further exploration consider the following

y = sin (at)

x = cos (bt ) for 0 t 2pi

Try different values of a = b and see what happens.

i. Do you still get a circle?

ii. If the answer to part (i) is yes, what is the radius of the circle?

iii. What is your conclusion when a = b?

CLICK here for GSP to explore more. Remember in this exercise we want

a =b all the time.

2. Now let us investigate situations when a and b are not the same. Refer to figure 4.

a = 2b

Figure 4

Can we say that the curve shown in figure 4 is a parabola? If we assume that this is a parabola, then it must be of the form . Compare the graph shown in figure 4 with that shown in figure 5.

Figure 5.

The graph shown in figure 5 is that of the function . Why are the two graphs the same?

We will still need to rely on our knowledge of trigonometry to answer this question?

Remember x = cos2t and y = sint. . We know that we can express cos 2t as .

We are looking for an equation of the form . First of all let us just write cos2t = (i).

Equation (i) can also be written as (ii). If equation (ii) is true then it follows that

. This is the same function that we used to graph the curve shown in figure. Now we know that curve in figure 4 is a parabola.

We will try one more time. We will still use a=2b, this time a=4 and b=2. In the first example we

used a=2 and b=1. Refer to figure 6.

Figure 5

The two graphs look alike. We seem to get a parabola whose axis of symmetry is in the x-axis and the vertex is at (1,0).

Can we generalize for x = cos at

y = sin bt

where a = 2b.

Use this LINK to explore that. You may also want to explore what happens when b = 2a.

Do you still get the same graph?

3. In this investigation we want to see what happens when we have the

following parametric equations.

x = a cos t

y = b sin t

We will consider cases when a is greater than b and also when b is greater

than a

First case: a > b. Refer to figure 5

Figure 5

We seem to get an ellipse here whose major axis is four units and the minor axis is 2 units.

Can you confirm that indeed this is an ellipse?

If it is an ellipse, what is the relationship between a, b and the lengths of the major and minor axes?

Hint: Remember the general equation for an ellipse . Express x = 2 cost as and express y = sint as . Use the trigonometry identity .

Second case b > a. Refer to figure 6

Figure 6

Now we get another ellipse with the major axis of length four units and the minor axis of length two units. When we compare the two curves it is very clear that the magnitude of a in comparison to b has an effect on the orientation of the ellipse.

We can conclude that that if a is greater than b, the major axis of the ellipse lies on the x-axis and if b is greater than a the major axis of the ellipse lies in the y-axis.

4. Our next investigation takes us to the parametric equations of the form

y = a cos t + h sin t

x = b sint t + h cost

a). We will use a =2 and b = 1 and -3 < h < 0

Figure 7

From our curves here we seem to get curves that look like stretched ellipses.

There are a number of things that we can observe here.

i. All the functions give curves that look like stretched ellipses, apart from the function where h = 0.

ii. h seems to determine the amount of stretch, the larger the h the greater the stretch of the curve.

iii. As the value of h decreases from 3 to 0, the angle of orientation of the curves is changing. For example the angle of inclination of the purple curve with respect to the x-axis is greater than the angle of the red curve.

Now let us look at the curves when 0 < h < 3

Figure 8

We can draw the same conclusions as in the first case. The only difference is the orientation of the curves.

Here is the final picture when all the curves are together. Refer to figure 7

Note: Curves whose functions are the same apart from the sign of h are drawn in the same color.

Figure 9

END