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Parametric Equations
Just like polar coordinates, parametric equations will
not show up on the Math IIC test very often. However, they do show
up occasionally, and knowing them might separate you from the pack.
Parametric equations are a useful way to express two variables
in terms of a third variable. The third variable is called the parameter.
Here is an example:
As the value of t changes, the ordered
pair (x, y) changes according
to the parametric equations, and a graph can be drawn.
Below is a graph of the parametric equations x =
3t – 2; y = –t +
4 for the range of values 0 ≤ t ≤ 3.
Eliminating the Parameter
As you might have guessed from the graph above, plotting
parametric equations by substituting values of the parameter can
be tedious. Luckily, some parametric equations can be reduced into
a single equation by eliminating the parameter. All this involves
is a little algebra.
Consider the parametric equations x =
2t; y = t + 1.
In the first equation, we can solve for t: t =
1/2 x.
Now we can substitute this value into the second equation to get y =
1/2 x +
1, which is a line we can easily sketch.
But be careful to keep the range of the original equations
in mind when you eliminate the parameter in parametric equations.
For example, by eliminating the parameter in the parametric equations x =
2t2; y =
4t2 + 3,
you arrive at the equation y = 2x +
3. The range of this function, however, does not include x values
below 0 or y values below 3 because
the ranges of the original parametric equations do not include these
values. So, the graph of these parametric equations actually looks
like the graph of y = 2x +
3 cut off below the point (0, 3):
When questions on parametric equations do appear on the
test, they’re usually quite simple. Given a parametric equation,
you should be able to recognize or sketch the proper graph, whether
by plotting a few points or by eliminating the parameter.
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