Parametric equations describe a curve by giving x and y as separate functions of a third variable, usually t. Instead of writing y directly in terms of x, you let a moving point trace a path as t changes. This is useful because many real paths, such as projectiles, circles, and moving vehicles, are easier to describe with time.
Parametric equations connect algebra, geometry, and motion in one flexible model.
The parameter t often represents time, but it can also be any variable that controls motion along a curve. A pair such as x = f(t) and y = g(t) tells you where the point is at each value of t. Sometimes you can eliminate t to find a rectangular equation in x and y, but the parametric form also contains direction, speed, and starting position information.
This makes parametric equations powerful for modeling curves that fail the vertical line test or have complicated motion.
Key Facts
- A parametric curve is defined by x = f(t) and y = g(t), where t is the parameter.
- To plot a parametric curve, choose t-values, compute x and y, then plot the ordered pairs (x, y).
- Eliminating the parameter means solving for t and substituting to get an equation involving only x and y.
- For x = a cos t and y = a sin t, eliminating t gives x^2 + y^2 = a^2.
- Velocity in parametric motion is v(t) = (dx/dt, dy/dt).
- The slope of a parametric curve is dy/dx = (dy/dt)/(dx/dt), when dx/dt is not 0.
Vocabulary
- Parameter
- A parameter is an independent variable that controls the values of other variables in a parametric equation.
- Parametric equations
- Parametric equations define coordinates such as x and y as functions of a separate variable, often written as t.
- Rectangular form
- Rectangular form is an equation involving only x and y, such as y = x^2 or x^2 + y^2 = 25.
- Elimination of the parameter
- Elimination of the parameter is the process of removing t to write a relationship directly between x and y.
- Parametric curve
- A parametric curve is the set of points traced by x = f(t) and y = g(t) as the parameter changes.
Common Mistakes to Avoid
- Ignoring the direction of increasing t is wrong because the same rectangular curve can be traced in different directions depending on the parametric equations.
- Treating t as if it must always be time is wrong because t is a parameter and only represents time when the context defines it that way.
- Eliminating t without checking restrictions is wrong because the parametric equations may cover only part of the rectangular curve.
- Using dy/dx = dy/dt directly is wrong because the slope requires dividing by dx/dt, so dy/dx = (dy/dt)/(dx/dt).
Practice Questions
- 1 For x = 2t + 1 and y = t - 3, find the points when t = 0, 1, and 2, then eliminate t to write y in terms of x.
- 2 For x = 3 cos t and y = 3 sin t, eliminate t to find the rectangular equation, and identify the shape of the curve.
- 3 Two parametric equations trace the same parabola: x = t, y = t^2 and x = 2s, y = 4s^2. Explain how the paths are related and whether the points are traced at the same rate.