Parametric equations describe both coordinates using a third variable, usually \(t\):
\[x=x(t),\qquad y=y(t).\]
They are especially useful for circles, ellipses, cycloids, loops, and curves that are difficult or impossible to express as a single function \(y=f(x)\).
1. Start with the unit circle
The standard parameterization
\[x(t)=\cos t,\qquad y(t)=\sin t,\qquad 0\le t\le2\pi\]
traces the unit circle once counterclockwise, beginning at \((1,0)\). In the parametric graph mode, enter cos(t) for \(x(t)\) and sin(t) for \(y(t)\), then set the parameter interval from \(0\) to \(2\pi\).
2. Understand the role of the parameter interval
The formulas alone do not determine how much of the curve appears. For the circle:
- \(0\le t\le\pi\) draws only the upper semicircle.
- \(-\pi/2\le t\le\pi/2\) draws the right semicircle.
- \(0\le t\le4\pi\) traces the same circle twice.
Choose the smallest interval that represents the intended curve without unnecessary repetition.
3. Graph an ellipse
An ellipse centered at \((h,k)\) with horizontal semiaxis \(a\) and vertical semiaxis \(b\) can be parameterized by
\[x(t)=h+a\cos t,\qquad y(t)=k+b\sin t.\]
For an ellipse centered at \((1,-2)\) with semiaxes 3 and 2, enter:
1+3*cos(t)-2+2*sin(t)
Use equal scale on the two axes when checking the shape. Unequal screen scaling can make a true circle look like an ellipse.
4. Trace direction and velocity
The parameter gives more information than the geometric path. The velocity vector is
\[\mathbf v(t)=\bigl(x'(t),y'(t)\bigr).\]
For the unit circle, \(\mathbf v(t)=(-\sin t,\cos t)\). At \(t=0\), the point is \((1,0)\) and the velocity points upward, confirming counterclockwise motion. Replacing \(\sin t\) with \(-\sin t\) reverses the direction.
5. Example: a Lissajous curve
Try
\[x(t)=\sin(3t),\qquad y(t)=\sin(2t),\qquad 0\le t\le2\pi.\]
Enter sin(3*t) and sin(2*t). The frequency ratio controls the number of lobes. Small changes to the coefficients can produce very different closed patterns.
6. Avoid missing detail
Curves with fast oscillation or sharp turns need enough sampling points. If a curve looks polygonal, has gaps, or cuts across a loop, reduce the parameter step or increase rendering quality if that option is available. Also verify that the parameter interval is expressed in radians unless the tool is explicitly set to degrees.
Common mistakes
- Entering both formulas into a standard \(y=f(x)\) field.
- Using degrees in formulas while the graph expects radians.
- Choosing an interval that draws only part of the curve.
- Using unequal axis scales and misreading the geometry.
- Ignoring the direction in which the curve is traced.
Open the Graphing Calculator and compare the circle, ellipse, and Lissajous examples. For another coordinate system, continue with polar equations and rose curves.