Finding Parametric Equations For A Graph

First Parametric equation

Second Parametric Equation

Rectangular Equation

x=4 t 2 −4

y = t

x=4 y 2 −4

x= t 2 −4

y= t 2

2y = t

x = (2y)2 - 4

x=4 y 2 −4

t

-2

-1

0

1

2

x = t + 2

0

1

2

3

4

y = t2 + 4t + 6

2

3

6

11

18

Step 1: Take the parameter equation and switch the roles of the parameter and the other variable.

Let t = x and rewrite the parameter equation by switching t and x.

t = 2 - x Original

x = 2 - t Switch t and x

Step 2: Substitute the expression for the variable in Step 1 into the rectangular equation

y= x 2 +1 Original

y= ( 2−t ) 2 +1 Substitute

y=( 4−4t+ t 2 )+1 Square

y=5−4t+ t 2 Add

Step 3: Sketch the curve.

List the two parametric equations and sketch a graph locating points at 0≤t≤3 and indicate the orientation of the curve.

Rectangular Equation: y= x 2 +1

Parametric Equations: x = 2 - t, y=5−4t+ t 2

Step 1: Take the parameter equation and switch the roles of the parameter and the other variable.

t= x Original

x= t Switch t and x

Step 2: Substitute the expression for the variable in Step 1 into the rectangular equation

y= 2 x 2 +1 Original

y= 2 t 2 +1 Substitute

y= 2 t+1 Square

Step 3: Sketch the curve.

List the two parametric equations and sketch a graph locating points at t = {0, 1, 2, 3, 4} and indicate the orientation of the curve.

Rectangular Equation: y= 2 x 2 +1

Parametric Equations: x= t , y= 2 t+1

Note that the domain of the graph is x≥0 because the domain of the parametric equation, t= x , limits x to values of zero or greater.