One curve has infinitely many parametrisations. x = t, y = t² and x = 2t, y = 4t² trace the same parabola at different speeds.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Orientation
Orientation is the direction the point moves as t increases. It is drawn as an arrow on the curve.
Reversing it
Replacing t with −t reverses the orientation while leaving the path untouched.
Restricted domains
Limiting t to an interval draws only part of the curve. With 0 ≤ t ≤ 180°, x = cos t and y = sin t give a half circle.
Speed
Points evenly spaced in t but far apart on the curve mean fast motion there. Closely spaced points mean slow motion.
A caution
Two parametrisations giving the same rectangular equation are not necessarily the same motion, and may not even cover the same part of the curve.
Same path, different journey
x = cos t, y = sin t traces the unit circle anticlockwise; x = cos(−t), y = sin(−t) traces the same circle clockwise. The curve is identical and the motion is not. Orientation is part of a parametrisation.
Mark the direction
A parametric sketch should carry arrows showing the direction of increasing t, and often the values of t at key points. Without them the sketch has thrown away half the information the equations contained.
Speed varies along the curve
Points equally spaced in t are not equally spaced along the path unless the speed is constant. Plotting at regular t intervals shows where the motion is fast and where it is slow, which a Cartesian graph cannot.
The interval matters
Restricting t to 0 ≤ t ≤ π traces only half the circle. Two parametrisations with the same formulas and different intervals describe different curves, so the interval must always be stated.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 4)
Step 3: Watch an Example
One step at a time.
Watch Ines Compare Two Parametrisations
Ines has x = t, y = t² and also x = −t, y = t².
- Step 1
She eliminates t in the first: y = x², a parabola.
Step 4: Your Turn
Practice makes it stick.
The Direction
Problem 1 of 2
x = −t, y = 3t. As t increases, does x increase or decrease? Enter 1 for increase, 2 for decrease.
The Half Circle
Problem 2 of 2
x = cos t, y = sin t with 0 ≤ t ≤ 180°. What fraction of the circle is drawn, as a decimal?
Same Path, Different Trip
1 of 8
x = t, y = t². What is y when t = −3?
2 of 8
x = 2t, y = 4t². Eliminating t gives y = xⁿ. What is n?
3 of 8
x = cos t, y = sin t with 0 ≤ t ≤ 90°. What fraction of the circle, as a decimal?
4 of 8
x = 3 − t, y = t. As t increases, does x increase or decrease? 1 or 2.
5 of 8
x = t², y = t. What is x when t = −4?
6 of 8
How many parametrisations does one curve have? Enter 0 for infinitely many, or the exact count.
7 of 8
Put the steps for analysing a parametric curve in order.
- 1Plot the points in increasing order of t.
- 2Draw an arrow showing the orientation.
- 3Eliminate t to check the rectangular equation.
- 4Build a table of t, x and y.
8 of 8
x = 5t, y = 5t. Eliminating t gives y = kx. What is k?
Step 5: Quick Check
Show what you know.
Question 1 of 2
x = −t, y = t². What is x when t = 6?
Question 2 of 2
Two parametrisations give the same rectangular equation. Are they the same motion?
What You Learned
- One curve has infinitely many parametrisations.
- Orientation is the direction of travel as t increases.
- Restricting the domain of t draws only part of the curve.