CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 10. Polar and Parametric Equations

Guidance

In your graphing calculator there is a parametric mode. Once you put your calculator into parametric mode, on the
graphing screen you will no longer seey= ,instead, you will see:

Notice how for plot one, the calculator is asking for two equations based on variableT:

x 1 T=f(t)

y 1 T=g(t)

In order to transform a parametric equation into a normal one, you need to do a process called “eliminating the
parameter.” To do this, you must solve thex=f(t)equation fort=f−^1 (x)and substitute this value oftinto they
equation. This will produce a normal function ofybased onx.
There are two major benefits of graphing in parametric form. First, it is straightforward to graph a portion of a
regular function using theTmin,TmaxandTste pin the window setting. Second, parametric form enables you to graph
projectiles in motion and see the effects of time.
Example A
Eliminate the parameter in the following equations.

x= 6 t− 2

y= 5 t^2 − 6 t

Solution:x= 6 t−2 Sox+ 62 =t. Now, substitute this value fortinto the second equation:

y= 5 (x+ 62 )^2 − 6 (x+ 62 )
Example B
For the given parametric equation, graph over each interval oft.
x=t^2 − 4
y= 2 t

a.− 2 ≤t≤ 0

b. 0≤t≤ 5

c.− 3 ≤t≤ 2

Solution:
a. A good place to start is to find the coordinates wheretindicates the graph will start and end. For− 2 ≤t≤ 0 ,t=
−2 andt=0 indicate that the points (0, -4) and (-4, 0) are the endpoints of the graph.