http://www.ck12.org Chapter 10. Polar and Parametric Equations
y= 3 ·cost
Solution: When parametric equations involve trigonometric functions you can use the Pythagorean Identity, sin^2 t+
cos^2 t=1. In this problem, sint=x 3 (from the first equation) and cost=y 3 (from the second equation). Substitute
these values into the Pythagorean Identity and you have:
(x
3) 2
+
(y
3) 2
= 1
x^2 +y^2 = 9This is a circle centered at the origin with radius 3.
Concept Problem Revisited
Parametric equations are often used when only a portion of a graph is useful. By limiting the domain oft, you
can graph the precise interval of the function you want. Parametric equations are also useful when two different
variables jointly depend on a third variable and you wish to look at the relationship between the two dependent
variables. This is very common in statistics where an underlying variable may actually be the cause of a problem
and the observer can only examine the relationship between the outcomes that they see. In the physical world,
parametric equations are exceptional at graphing position over time because the horizontal and vertical vectors of
objects in free motion are each dependent on time, yet independent of one another.
Vocabulary
“Eliminating the parameter”is a phrase that means to turn a parametric equation that hasx=f(t)andy=g(t)into
just a relationship betweenyandx.
Parametric formrefers to a relationship that includesx=f(t)andy=g(t).Parameterizationalso means parametric
form.
Guided Practice
- Find the parameterization for the line segment connecting the points (1, 3) and (4, 8).
- A tortoise and a hare start 202 feet apart and then race to a flag halfway between them. The hare decides to take
a nap and give the tortoise a 21 second head start. The hare runs at 9.8 feet per second and the tortoise hustles along
at 3.2 feet per second. Who wins this epic race and by how much?