Parametric Equations

Suppose that we have an equation representing y as a function of x. If the values of both x and y change with respect to time over a given interval of time, we can introduce a third variable, t, equations relating x and t and y and t, and an interval for t. These equations are called parametric equations and t is called the parameter. The set of points (x, y) obtained for different values of t over the given interval form the graph of the parametric equations. The parametric equations and the graph form what is called a plane curve. The plane curve is graphed in a specific orientation determined as the value of the parameter increases from the beginning value in the interval to the ending value in the interval.

Consider the following parametric equations: x t 2, y  t2  1,  2  t  3

In order to graph the plane curve, select values for the parameter t and make a table of values. Plot the points in the table of values and trace the path of the curve, drawing arrows to indicate the orientation or direction of the graph as it is drawn.

For the above parametric equations, we obtain the following table of (x, y) values:

t -2 -1 0 1 2 3 x -4 -3 -2 -1 0 1 y 5 2 1 2 5 10 To graph the plane curve using a TI-84+ graphing calculator. Press , move the cursor to the fourth row, and then over to PAR, and press  as shown on the screen on the left below. Press  to set up the values for t, x, and y, based on the table of values shown above (see the two screen shown below). Press  and enter the parametric equations (press  to enter t).

Press  to display the graph as shown below. Press  and the right arrow key to trace through the path of the graph along the points given in the table.

As shown above, the path of the plane curve starts at the point (–4, 5), then moves down to the points (–3, 2) and (–2, 1), and then proceeds up to the points (–1, 2), (0, 5), and (1, 10).

We can find the rectangular equation that represents the set of parametric equations through a process called eliminating the parameter. In order to do this, solve for t in one of the parametric equation and substitute into the other parametric equation.

In our example above, it would be easiest to solve the first equation for t giving t = x + 2. Substituting into the second equation, we obtain the following: x t 2, y  t2  1,  2  t  3 t x 2  y  x  22  1

In this case we can graph the resulting rectangular equation to check our answer as shown below.

Homework – Sketch the curve represented by the parametric equations (draw arrows to indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter.

1.x t2  1, y  t  3,  2  t  3 2.x 3 t  1, y  t  2,  1  t  4 3.x t2  t , y  t  1,  2  t  3 4.x t  1, y  t  2, 0  t  5 5.x t  2, y  t  1 , 1  t  6