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.
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.