Parametric equations describe a curve by letting both coordinates depend on a third variable, usually called the parameter t. Instead of writing y directly as a function of x, we write x = f(t) and y = g(t). This is useful when a point moves through the plane, because the parameter can represent time.
The same curve can be traced in different directions or at different speeds depending on how x and y change with t.
A parametric curve is studied by plotting points for increasing values of t and connecting them in order. Eliminating the parameter can sometimes give a familiar Cartesian equation, but it may lose important information about direction, timing, or restricted portions of the curve. Calculus adds tools for finding slope, speed, and acceleration from the component functions.
For example, dy/dx = (dy/dt)/(dx/dt) shows how the curve rises or falls as the moving point travels along its path.
Understanding Calculus: Parametric Equations and Curves
The parameter gives each point on the curve a history. Two points may occupy the same location at different parameter values, which means the path crosses itself or returns to an earlier place. A Cartesian equation usually cannot show this history.
For instance, a moving object can travel around a circle twice, pause at one point, or reverse direction. Its geometric track may look unchanged, but its parametric description records the actual motion. This matters in animation, robotics, GPS tracking, and physics, where position alone is not enough to describe what an object is doing.
The derivative of each coordinate describes one part of the motion. The derivative of x with respect to t tells how quickly the point moves left or right. The derivative of y with respect to t tells how quickly it moves down or up.
Together, these two rates form a velocity vector. A large horizontal rate does not necessarily mean a large overall speed, because the vertical rate matters too. Speed comes from the length of the velocity vector.
If both coordinate rates are zero, the point is momentarily at rest. The curve may still have an interesting shape there, such as a sharp point, a loop, or a change in direction.
Slope needs careful attention because the parameter is not the same as x. To find the slope of the path, divide the rate of change of y by the rate of change of x. This is an application of the chain rule.
When the horizontal rate is zero while the vertical rate is not zero, the curve has a vertical tangent. In that situation, ordinary slope is undefined, but the parametric equations still describe the motion clearly.
When both rates are zero, the usual slope rule gives zero divided by zero, so it cannot decide the tangent line. Students should examine nearby parameter values or use higher derivatives to understand the shape at that point.
A second derivative can reveal how the curve bends. First find the slope as a function of the parameter. Then differentiate that slope with respect to the parameter and divide by the derivative of x with respect to the parameter.
This gives the second derivative of y with respect to x. Its sign can indicate whether the path bends upward or downward, as long as the calculation is defined. In motion problems, differentiating the velocity components gives acceleration components.
Acceleration can point in a different direction from velocity, which happens when a car turns at nearly constant speed. When graphing, always state the parameter interval, mark a few values in order, and check for repeated points, endpoints, vertical tangents, and places where the point stops.
Key Facts
- A parametric curve is given by x = f(t), y = g(t), where t is the parameter.
- To trace a curve, compute ordered pairs (x(t), y(t)) for increasing values of t.
- If dx/dt is not 0, then dy/dx = (dy/dt)/(dx/dt).
- The speed of a moving point is v = sqrt((dx/dt)^2 + (dy/dt)^2).
- For x = r cos t and y = r sin t, eliminating t gives x^2 + y^2 = r^2.
- Eliminating the parameter may remove direction of motion and restrictions on t.
Vocabulary
- Parameter
- A parameter is an independent variable, often t, that controls both coordinates of a point on a curve.
- Parametric equations
- Parametric equations are equations such as x = f(t) and y = g(t) that define a curve using a shared parameter.
- Parametric curve
- A parametric curve is the set of points traced by (x(t), y(t)) as the parameter varies.
- Eliminating the parameter
- Eliminating the parameter means rewriting parametric equations as a relationship between x and y only.
- Speed
- Speed is the rate at which a point moves along a parametric curve, given by sqrt((dx/dt)^2 + (dy/dt)^2).
Common Mistakes to Avoid
- Treating t as the y-value is wrong because t is a separate input that determines both x and y.
- Plotting points but not connecting them in order is wrong because the order of increasing t shows the direction of motion.
- Using dy/dx = dy/dt divided by x is wrong because the correct formula is dy/dx = (dy/dt)/(dx/dt), when dx/dt is not 0.
- Eliminating the parameter and ignoring restrictions is wrong because the Cartesian equation may include points that the original parameter range never reaches.
Practice Questions
- 1 For x = 2t + 1 and y = t^2, find the points on the curve when t = -1, 0, and 2.
- 2 For x = 3 cos t and y = 3 sin t, eliminate the parameter and find the speed when t = pi/4.
- 3 Two parametrizations trace the same circle: x = cos t, y = sin t and x = cos(2t), y = sin(2t). Explain how their paths are the same but their motion is different.