Parametric Equations Grapher

About the Parametric Equations Grapher

Plot parametric curves by entering x(t) and y(t) separated by a semicolon (t is entered as x).

How to use this calculator

1. Enter the x-coordinate formula in terms of x, which represents the parameter t, for example 2*cos(x).
2. Type a semicolon, then enter the y-coordinate formula, for example 2*sin(x).
3. If needed, include a domain for the parameter so the graph shows the correct part of the curve, such as from 0 to 2*pi.
4. Plot the curve and check that the x and y expressions work together to trace the path you want.

The formula explained

A parametric graph uses two formulas, x(t) and y(t), to compute the coordinates of points on a curve. This tool plots the path traced as the parameter changes.

• x = the parameter t, entered in the input box as x
• y = the y-coordinate formula, written as a function of the parameter
• t = the parameter that controls the point moving along the curve

Step by step method

1. Choose a parameter value, such as t = 0, 1, 2, and so on.
2. Compute x(t) and y(t) for each chosen value.
3. Plot each point, then connect them in the order the parameter increases.

Worked example

Problem. Graph the circle of radius 3 centered at the origin using parametric equations.

1. Write the equations as x(t) = 3 cos(t) and y(t) = 3 sin(t).
2. Use values from 0 to 2pi so the full circle is traced.
3. When t = 0, the point is (3, 0), and as t increases the points trace a circle counterclockwise.

Tips and common mistakes

• Remember that the x box stands for the parameter t here, not the usual x-coordinate, so write formulas in terms of x as the parameter.
• If the graph looks incomplete, check the parameter range, because many curves need a specific interval to show the full shape.

Frequently asked questions