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: Picture It
Start with something you already know.
A walk logged by the minute
- A phone logs your walk once a minute.
- Each entry holds the minute, how far east, and how far north.
- Drawn on a map, the entries make a trail.
- The trail shows where you went, but not when, or which way.
- Read the log in order, and the direction comes back.
Parametric equations are that log: x and y each written against a third variable, t.
Eliminating t draws the trail as one equation, and the times and the direction are lost.
The trail’s steepness is the north speed over the east speed. Unlike a log, a curve has a point at every t, not only each minute.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A point moves so that at time t it sits at x = t + 1, y = 2t + 3. Work out where it is at t = 0, 1, 2 and 3. The four points lie on one straight line. What is the slope of that line?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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 3: 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 4: 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 5: Watch an Example
One step at a time.
Watch Noor Follow x = 2t, y = t² + 1
A fresh path: a table of points, the parameter eliminated, the slope as a ratio, a check against the Cartesian form, and the one place the path runs flat.
- Step 1
Noor starts with a table. t = −1, 0, 1, 2 give (−2, 2), (0, 1), (2, 2) and (4, 5). Read in order, the point comes down to (0, 1) and then climbs away to the right, so that is the direction of travel.
Watch Leo Find a Tangent Line, Then Finish It Yourself
The path x = t² + 1, y = t³. Leo finds the slope in terms of t. The slope at t = 2, the tangent line there, and a second visit to the same x are yours.
- Step 1
Leo differentiates each coordinate against t: dx/dt = 2t and dy/dt = 3t².
Step 6: Your Turn
Practice makes it stick.
Where the Ball Lands
Problem 1 of 16
A ball is kicked from the ground. t seconds later it is x = 10t meters along the field and y = 20t − 5t² meters up. How far along the field does it land, in meters?
meters
Revenue Against Advertising
Problem 2 of 16
A small shop tracks month t of the year. By month t it has spent x = 200t + 100 dollars on advertising in total, and taken y = 50t² + 1000 dollars in revenue in total. At t = 4, how many dollars is total revenue rising for each dollar that total advertising rises?
dollars per dollar
Point, Path and Slope
Each curve is two equations in t. Decide first whether the question wants a point, the path, or a slope.
Problem 3 of 16
x = 2t − 1, y = t² + 3. What is y at the moment when x = 5?
Problem 4 of 16
x = t − 2, y = t² + 1. Eliminate the parameter and type y in terms of x.
Problem 5 of 16
x = t² + 2t, y = t³ − 4. What is the slope dy/dx at t = 1? Give it as a decimal.
Problem 6 of 16
x = 3t + 1, y = t² − 6t. At what value of t is the tangent horizontal?
Problem 7 of 16
x = t² − 4t, y = t³. The tangent is vertical at one point. What is that point’s x-coordinate?
Whose Step?
One line on each page is a slip. Find it, then put it right.
Problem 8 of 16
Ravi found the slope of x = t³, y = 12t at t = 1. One line is a slip. Which?
Ravi’s slope of x = t³, y = 12t at t = 1
- 1dx/dt = 3t², dy/dt = 12
- 2at t = 1: dx/dt = 3, dy/dt = 12
- 3dy/dx = 3 ÷ 12
- Answerslope = 0.25
Problem 9 of 16
Worked correctly, what is the slope of x = t³, y = 12t at t = 1?
Problem 10 of 16
Sam eliminated the parameter from x = 2t + 1, y = t². One line is a slip. Which?
Sam’s Cartesian form of x = 2t + 1, y = t²
- 1x = 2t + 1
- 2t = (x + 1)/2
- 3y = ((x + 1)/2)²
- Answery = (x + 1)²/4
Problem 11 of 16
Done correctly, what is the Cartesian form of x = 2t + 1, y = t²? Type y in terms of x.
Problem 12 of 16
Ana wrote the tangent line to x = t + 3, y = t² at t = 1. One line is a slip. Which?
Ana’s tangent line to x = t + 3, y = t² at t = 1
- 1dx/dt = 1, dy/dt = 2t, so dy/dx = 2t
- 2at t = 1: slope 2
- 3at t = 1: x = 1, y = 1
- Answery − 1 = 2(x − 1), so y = 2x − 1
Problem 13 of 16
Done correctly, what is the tangent line to x = t + 3, y = t² at t = 1? Type it solved for y.
Pick the Move
Choose the move. Neither needs working all the way through.
Problem 14 of 16
x = 5 cos t, y = 5 sin t. Which move removes t most cleanly?
Problem 15 of 16
For x = t³ − 3t, y = t², dx/dt = 3t² − 3 is 0 at t = 1. What must be checked before calling the tangent there vertical?
Same Curve?
Eliminate t in your head. Select every entry that fits.
Problem 16 of 16
Select every pair of equations that traces the whole parabola y = x².
Step 7: Quick Check
Show what you know.
Question 1 of 1
dx/dt = 3, dy/dt = 15. What is dy/dx?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
x = 3t, y = t³ + kt. For which number k is the tangent horizontal at t = 2?
Stretch 2 of 2
The path x = t², y = t³ − 4t crosses itself at one point: two different times give the same (x, y). Find that point. Two tangent lines pass through it; what is the positive one of their two slopes?
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.
For the grown-up
The misconception this lesson targets is that a parametric curve is just y = f(x) written the long way: that dy/dt is the slope, that the Cartesian equation tells the whole story, and that the value of t can stand in for x. A student who holds it divides the rates upside down, or not at all, writes a tangent line through (t, t²) instead of (x, y), and is surprised when one curve shows two slopes above the same x. The habit that fixes it is to say the roles out loud: t is the clock, x and y are both read off the clock, the slope is the y-rate over the x-rate, and the point always comes from putting t into both equations.