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

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

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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 .
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Questions & Answers

can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
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
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
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
Practice Key Terms 1

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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