<< Chapter < Page Chapter >> Page >
Graph of a circle in the rectangular coordinate system - the vertical line test shows that the circle r^2 = x^2 + y^2 is not a function. The dotted red vertical line intersects the function in two places - it should only intersect in one place to be a function.

However, if we were to graph each equation on its own, each one would pass the vertical line test and therefore would represent a function. In some instances, the concept of breaking up the equation for a circle into two functions is similar to the concept of creating parametric equations, as we use two functions to produce a non-function. This will become clearer as we move forward.

Parametric equations

Suppose t is a number on an interval, I . The set of ordered pairs, ( x ( t ) , y ( t ) ) , where x = f ( t ) and y = g ( t ) , forms a plane curve based on the parameter t . The equations x = f ( t ) and y = g ( t ) are the parametric equations.

Parameterizing a curve

Parameterize the curve y = x 2 1 letting x ( t ) = t . Graph both equations.

If x ( t ) = t , then to find y ( t ) we replace the variable x with the expression given in x ( t ) . In other words, y ( t ) = t 2 1. Make a table of values similar to [link] , and sketch the graph.

t x ( t ) y ( t )
4 4 y ( 4 ) = ( 4 ) 2 1 = 15
3 3 y ( 3 ) = ( 3 ) 2 1 = 8
2 2 y ( 2 ) = ( 2 ) 2 1 = 3
1 1 y ( 1 ) = ( 1 ) 2 1 = 0
0 0 y ( 0 ) = ( 0 ) 2 1 = 1
1 1 y ( 1 ) = ( 1 ) 2 1 = 0
2 2 y ( 2 ) = ( 2 ) 2 1 = 3
3 3 y ( 3 ) = ( 3 ) 2 1 = 8
4 4 y ( 4 ) = ( 4 ) 2 1 = 15

See the graphs in [link] . It may be helpful to use the TRACE feature of a graphing calculator to see how the points are generated as t increases.

Graph of a parabola in two forms: a parametric equation and rectangular coordinates. It is the same function, just different ways of writing it.
(a) Parametric y ( t ) = t 2 1 (b) Rectangular y = x 2 1
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Construct a table of values and plot the parametric equations: x ( t ) = t 3 , y ( t ) = 2 t + 4 ; 1 t 2.

t x ( t ) y ( t )
1 4 2
0 3 4
1 2 6
2 1 8
Got questions? Get instant answers now!

Finding a pair of parametric equations

Find a pair of parametric equations that models the graph of y = 1 x 2 , using the parameter x ( t ) = t . Plot some points and sketch the graph.

If x ( t ) = t and we substitute t for x into the y equation, then y ( t ) = 1 t 2 . Our pair of parametric equations is

x ( t ) = t y ( t ) = 1 t 2

To graph the equations, first we construct a table of values like that in [link] . We can choose values around t = 0 , from t = 3 to t = 3. The values in the x ( t ) column will be the same as those in the t column because x ( t ) = t . Calculate values for the column y ( t ) .

t x ( t ) = t y ( t ) = 1 t 2
3 3 y ( 3 ) = 1 ( 3 ) 2 = 8
2 2 y ( 2 ) = 1 ( 2 ) 2 = 3
1 1 y ( 1 ) = 1 ( 1 ) 2 = 0
0 0 y ( 0 ) = 1 0 = 1
1 1 y ( 1 ) = 1 ( 1 ) 2 = 0
2 2 y ( 2 ) = 1 ( 2 ) 2 = 3
3 3 y ( 3 ) = 1 ( 3 ) 2 = 8

The graph of y = 1 t 2 is a parabola facing downward, as shown in [link] . We have mapped the curve over the interval [ −3 , 3 ] , shown as a solid line with arrows indicating the orientation of the curve according to t . Orientation refers to the path traced along the curve in terms of increasing values of t . As this parabola is symmetric with respect to the line x = 0 , the values of x are reflected across the y -axis.

Graph of given downward facing parabola.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Parameterize the curve given by x = y 3 2 y .

x ( t ) = t 3 2 t y ( t ) = t

Got questions? Get instant answers now!

Finding parametric equations that model given criteria

An object travels at a steady rate along a straight path ( −5 , 3 ) to ( 3 , −1 ) in the same plane in four seconds. The coordinates are measured in meters. Find parametric equations for the position of the object.

The parametric equations are simple linear expressions, but we need to view this problem in a step-by-step fashion. The x -value of the object starts at −5 meters and goes to 3 meters. This means the distance x has changed by 8 meters in 4 seconds, which is a rate of 8 m 4  s , or 2 m / s . We can write the x -coordinate as a linear function with respect to time as x ( t ) = 2 t 5. In the linear function template y = m x + b , 2 t = m x and 5 = b .

Similarly, the y -value of the object starts at 3 and goes to −1 , which is a change in the distance y of −4 meters in 4 seconds, which is a rate of 4  m 4  s , or 1 m / s . We can also write the y -coordinate as the linear function y ( t ) = t + 3. Together, these are the parametric equations for the position of the object, where x and y are expressed in meters and t represents time:

x ( t ) = 2 t 5 y ( t ) = t + 3

Using these equations, we can build a table of values for t , x , and y (see [link] ). In this example, we limited values of t to non-negative numbers. In general, any value of t can be used.

t x ( t ) = 2 t 5 y ( t ) = t + 3
0 x = 2 ( 0 ) 5 = 5 y = ( 0 ) + 3 = 3
1 x = 2 ( 1 ) 5 = 3 y = ( 1 ) + 3 = 2
2 x = 2 ( 2 ) 5 = 1 y = ( 2 ) + 3 = 1
3 x = 2 ( 3 ) 5 = 1 y = ( 3 ) + 3 = 0
4 x = 2 ( 4 ) 5 = 3 y = ( 4 ) + 3 = 1

From this table, we can create three graphs, as shown in [link] .

Three graphs side by side. (A) has the horizontal position over time, (B) has the vertical position over time, and (C) has the position of the object in the plane at time t. See caption for more information.
(a) A graph of x vs. t , representing the horizontal position over time. (b) A graph of y vs. t , representing the vertical position over time. (c) A graph of y vs. x , representing the position of the object in the plane at time t .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply
Practice Key Terms 1

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask