Parametric equations of curves
Describe a curve by giving x and y separately in terms of a parameter, plot it from a table of parameter values, and convert to Cartesian form by eliminating the parameter.
What a learner can do afterwards
- Plot the curve x = 2t, y = t² for a range of t values
- Eliminate t to get the Cartesian equation of that curve
- Use sin²t + cos²t = 1 to show that x = 3 cos t, y = 3 sin t traces a circle for 0 <= t <= 2pi radians
1 · Read
A parameter is an input used to calculate both coordinates. It may represent time or an angle. For x = 2t and y = t², use the same allowed t in both equations to get one point (x, y). Make a table with columns for t, x and y. Plot each pair with x horizontal and y vertical. For these smooth curves, sketch a smooth path through the points in increasing t order.
Use each t in both equations, then plot each (x, y) on standard axes. From the origin, move horizontally by x, then vertically by y. Positive means right or up; negative means left or down. Here (-4, 4) is 4 units left and 4 up, while (4, 4) is 4 right and 4 up. The points lie on an upward U whose lowest point is (0, 0). Sketch a smooth curve through them. As t increases, add arrows down the left branch to the origin, then up the right branch.
These cells show table entries, not a coordinate graph. Plot each pair on coordinate axes.
Eliminating t gives a relation between x and y without t. For x = 2t and y = t², solve t = x/2, then substitute: y = (x/2)² = x²/4. With x = t + 1 and y = t², use t = x - 1 to get y = (x - 1)². Expand by multiplying each term in one bracket by each term in the other: x² - x - x + 1 = x² - 2x + 1. To find y at a known x, solve for t first, then use that same t in the y equation.
Keep the parameter range when eliminating t. For x = 2t, y = t² and -1 <= t <= 3, the endpoints give x = -2 and x = 6. Since x increases with t, every x between them is reached. The Cartesian description is y = x²/4 with -2 <= x <= 6. Without this restriction, the equation includes points outside the traced part of the parabola.
For x = r cos t and y = r sin t, where r > 0, use cos²t + sin²t = 1: x² + y² = r². This is a circle of radius r centred at (0, 0). Angles here are in radians; pi/2 is a quarter turn, or 90 degrees. At t = 0, cos t = 1 and sin t = 0. At t = pi/2, cos t = 0 and sin t = 1. For 0 <= t <= 2pi, all angles of one turn are covered, so the whole circle is traced once.
Arrows on a sketch show travel as t increases. On standard axes, x = r cos t and y = r sin t, with r > 0, move anticlockwise from (r, 0) when t starts at 0. Changing y to -r sin t gives (r, 0) at t = 0 and (0, -r) at t = pi/2, so travel is clockwise. For 0 <= t <= 2pi, both versions trace the same set of points. They are different parametrisations of the same circle.
An angle range can restrict a circle to an arc. For x = r cos t and y = r sin t, with r > 0 and 0 <= t <= pi, sine is nonnegative, so y >= 0. The point travels from (r, 0) through (0, r) to (-r, 0). It traces the whole upper semicircle, including both endpoints. Its Cartesian description is x² + y² = r² with y >= 0.
Use the same t for both coordinates. Plot in increasing t order, eliminate t, and keep the restrictions from its allowed range.
The lesson and questions were written for this dataset. OpenStax and Khan Academy records informed the mathematical checks and teaching design; their prose is not reproduced.
The three readings, six questions and revised full-circle flashcard explanation were written for this dataset. OpenStax, Khan Academy and WeBWorK records informed the design. The two linked videos are credited to Khan Academy.
More to read
One input, two coordinates
Suppose a point follows x = 5t and y = t². At t = -2, its coordinates are x = -10 and y = 4. Both calculations use the same input, so the position is (-10, 4).
To prepare a sketch, make a table with columns for t, x and y. Put the rows in increasing t order. Plot the coordinate pairs on axes with x horizontal and y vertical. For these equations, draw a smooth curve through the plotted positions. Arrows show the travel as t increases.
A table is a sample of positions. Its rows do not turn a smooth curve into straight segments. More rows can help you sketch between widely separated positions. Keep the parameter values beside the coordinates so you can tell when each sampled position is reached.
Keep the allowed part of the curve
Take x = t + 3 and y = t², with 0 <= t <= 4. The x equation gives t = x - 3. Replacing t in the y equation gives y = (x - 3)². The brackets keep the whole replacement inside the square.
Now translate the allowed inputs. At t = 0, x = 3. At t = 4, x = 7. As t increases between those bounds, x reaches every value from 3 to 7. The Cartesian description is y = (x - 3)² with 3 <= x <= 7.
Without the restriction, the equation also includes points outside the traced part of the parabola. Eliminating t does not remove the original limits. It gives a relation between x and y, which need not always be written as y in terms of x.
The path and the direction of travel
For x = 2 cos t and y = 2 sin t, squaring and adding gives x² + y² = 4(cos²t + sin²t) = 4. Each position lies on the circle of radius 2 centred at the origin.
Measure t in radians and let it increase from 0 to 2pi. The point travels anticlockwise from (2, 0), through (0, 2) and (-2, 0), then returns to (2, 0) after one turn.
If y becomes -2 sin t, the point instead travels clockwise. Over the same full-turn interval it visits the same circle. The parametrisation changes, while the set of points stays the same.
A shorter interval can change the point set. With the original positive-sine equations and 0 <= t <= pi, only the upper semicircle is reached. Its Cartesian description needs both x² + y² = 4 and y >= 0.
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