A parametric curve gives x and y each as a function of a third variable, usually time. It can double back or cross itself, which y = f(x) never can. Eliminating t recovers a Cartesian relation but loses the direction. The slope is dy/dx = (dy/dt)/(dx/dt), a ratio of two rates.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
A table first
t = −1, 0, 1, 2 gives (0, 1), (1, 0), (2, 1), (3, 4). Plotted in order, the point comes down to the vertex and climbs away to the right. The order of the points is the direction of travel.
Eliminating the parameter
Solve the simpler equation for t and substitute. For x = 3 cos t, y = 3 sin t, square and add instead: x² + y² = 9. The identity removes t in one move.
What is lost
The Cartesian form forgets the direction and any restriction on t. With 0 ≤ t ≤ π the circle above is only its upper half, but x² + y² = 9 shows the whole thing.
Why the slope is a ratio
The chain rule gives dy/dt = (dy/dx)(dx/dt). Divide by dx/dt and dy/dx appears without ever eliminating t. A slope is a rise over a run, and both are now rates in t.
Vertical tangents
Where dx/dt = 0 and dy/dt ≠ 0 the tangent is vertical and the slope formula does not apply. For the circle: dx/dt = −3 sin t = 0 at t = 0 and π, the two ends of the horizontal diameter.
A tangent line
At t = 2 the point is (3, 4) and the slope is 4: y − 4 = 4(x − 3). The point comes from the parametric equations, the slope from the ratio. Both are needed.
Step 2: Try It Yourself
Tap and try it out.
x = 3 cos t, y = 3 sin t. Find the Cartesian form, dy/dx, the slope at t = π/6, and where the tangent is vertical
- 1x² + y² = 9 cos²t + 9 sin²t = 9a circle of radius 3, counterclockwise
Step 0 of 4
Step 3: In Real Life
A drone’s log
A drone logs x and y against time: two functions of a third variable. Eliminate t for the path; keep it for the timing. The slope along the path comes from the two rates.
Step 4: Watch an Example
One step at a time.
Watch Omar Eliminate the Parameter
Omar converts x = t + 1, y = t² to Cartesian form.
- Step 1
He solves the first for t: t = x − 1, and substitutes into the second: y = (x − 1)².
Step 5: Your Turn
Practice makes it stick.
At a Moment
Problem 1 of 2
x = 2t, y = t². What is x when t = 3?
The Slope
Problem 2 of 2
dx/dt = 2 and dy/dt = 6. What is dy/dx?
Parametrically
1 of 8
x = 3t, y = t. What is x when t = 4?
2 of 8
dx/dt = 4, dy/dt = 12. What is dy/dx?
3 of 8
dx/dt = 5, dy/dt = 0. What is dy/dx?
4 of 8
x = t, y = t². Eliminating t gives y = x to what power?
5 of 8
Select every true statement about parametric curves.
6 of 8
x = t + 2, y = 3t. What is y when x = 5?
7 of 8
dx/dt = 0 at a point. Is the tangent vertical there?
8 of 8
Order the steps for eliminating the parameter.
- 1Substitute into the other
- 2Simplify to a relation in x and y
- 3Solve one equation for t
Step 6: Quick Check
Show what you know.
Question 1 of 1
dx/dt = 3, dy/dt = 15. What is dy/dx?
What You Learned
- A parametric curve gives x and y each as a function of a parameter.
- Eliminating the parameter recovers a Cartesian relation but loses direction.
- dy/dx is the ratio (dy/dt) / (dx/dt), and the tangent is vertical where dx/dt = 0.
- A tangent line needs the point from the parametric equations and the slope from the ratio.