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.
Understanding Math: Parametric Equations
A parametric model does more than draw a shape. It gives a timeline for the point's location. Two different parameter values can produce the same location, which means the point may return to a place it visited before.
This happens in loops, repeated paths, and oscillations. The order of the parameter values tells the direction of travel.
A graph of the curve alone cannot always show that direction. Marking a few points in increasing parameter order helps reveal how the path is traced.
A table is often the safest starting method. Choose parameter values that are easy to calculate, then record the horizontal coordinate and the vertical coordinate for each one. Include negative values when the parameter range allows them.
Plot the points in their time order rather than simply connecting nearby points. A poor choice of values can hide a sharp turn, a loop, or a repeated section.
Smaller steps are needed where the coordinates change quickly. Larger steps may be enough along a nearly straight part of the path.
Rates of change explain the motion on the curve. The horizontal rate tells how fast the point moves left or right. The vertical rate tells how fast it moves down or up.
Together, these rates form the velocity. A positive horizontal rate means motion to the right, while a negative one means motion to the left. The slope of the path comes from dividing the vertical rate by the horizontal rate.
When the horizontal rate is zero and the vertical rate is not zero, the curve has a vertical tangent. When the vertical rate is zero and the horizontal rate is not zero, it has a horizontal tangent. These cases are important because direct division would fail at a vertical tangent.
Acceleration describes how velocity changes. It can make a moving object speed up, slow down, or turn. For a thrown ball, gravity causes a steady downward change in the vertical velocity.
The horizontal velocity may stay constant if air resistance is ignored. This produces a curved path even though no horizontal force is included in the simple model.
In animation, games, and computer graphics, the same ideas control an object's position at each moment. Engineers use related models when planning machine parts, vehicle routes, and robot movements.
Removing the parameter can identify the overall geometric curve, but it can remove useful information. A rectangular equation may show a circle, for example, yet it does not show where the motion began, whether it goes clockwise, or how fast the point moves. It may even include parts that the parameter never reaches because of restrictions on its values.
When working problems, keep the allowed parameter interval beside the equations. Check the starting point, ending point, direction, repeated points, and any places where a rate becomes zero. These details turn an equation into a complete description of 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.