1998 AB Exam

1.4 Parametric Equations

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Greg Kelly, 2005

Mt. Washington Cog Railway, NH

There are times when we need to describe motion (or a curve) that is not a function.

We can do this by writing equations for the x and y coordinates in terms of a third variable (usually t or ).

x f t y g t These are calledparametric equations.

“t” is the parameter. (It is also the independent variable)

Example 1: 0x t y t t

To graph on the TI-nspire:

Press menu 3

Graph Entry/Edit

3

Parametric

Input formulas for x and y, the range for t, and the size of the step between points.

enterPress

(You will need to use the delete key.)

(Your viewing window will probably be different.)

We can confirm this algebraically:

x t y t

x y

2x y 0x

2y x 0x

parabolic function

Press menu

4

Window / Zoom

5

Zoom - Standard

t

Circle:

If we let t = the angle, then:

cos sin 0 2x t y t t

Since: 2 2sin cos 1t t

2 2 1y x

2 2 1x y We could identify the parametric equations as a circle.

2

1

Window Settings

Graph on your calculator:

xt1 cos( )tyt1 sin( )t

menu 3 3

To find the trig functions, use the key.trig

Change the window settings. menu 4

Now square it up. menu 4

Zoom - Square

B

5

Trace

You can watch the direction and relative velocity of the graph by using the trace function:

menu 1

Graph Trace

Notice the x, y and t values displayed.

Use the right and left arrow keys to watch the position change as t changes.

You can enter a specific value for t, like .

/ 2

Holding a key down makes the motion continuous.

Change the speed by changing the size of the steps:

5

Trace

menu 3

Trace Step…

Smaller steps slow the graph down.

The TI-nspire can also graph conic sections directly without converting to parametric equations.

To clear the screen, press menu

1

Actions

4

Delete all

enter

Now we can enter the Cartesian equation for a circle.

The TI-nspire can also graph conic sections directly without converting to parametric equations.

menu 3

Graph Entry/Edit

2

Equation

3

Circle

1

2 2 2x h y k r

The horizontal and vertical shifts are zero, and the radius is 1.

Ellipse: 3cos 4sinx t y t

cos sin3 4

x yt t

2 22 2cos sin

3 4

x yt t

2 2

13 4

x y

This is the equation of an ellipse.

Converting Between Parametric and Cartesian Equations

We have seen two techniques for converting from parametric to Cartesian:

The first method is called eliminating the parameter. It requires solving one equation for t and substituting into the other equation to eliminate t. This is possible when the graph is a function.

2 2sin cos 1t t The second method used the Pythagorean identity to eliminate t by using the fact that .

Both of these methods only work sometimes. There are many curves that can only be described parametrically.

On the other hand, changing from the Cartesian equation for a function to a parametric equation always works and it is easy!

The steps are: 1) Replace x with t in the original equation.

2) Let x = t .

Example: 32 5 4y x x

32 5 4y t t x t

becomes:

In the special case where we want the parametrization for a line segment between two points, we could find the Cartesian equation first and then convert it to parametric, but there is an easier way. We will use an example to illustrate:

Find a parametrization for the line segment with endpoints(-2,1) and (3,5).

Using the first point, start with: 2x at 1y bt

Notice that when t = 0 you get the point (-2,1) .

Substitute in (3,5) and t = 1 . 3 2 1a 5 1 1b 5 a 4 b

The equations become: 2 5x t 1 4y t 0 1t