Parametric Equations — Complete Guide
Parametric form, eliminating the parameter, orientation, common curve types, projectile motion, and real-world applications — with three elimination methods worked step-by-step.
What Are Parametric Equations?
Instead of writing y as a function of x, parametric equations express both x and y as functions of a third variable — the parameter, usually called t.
Why Use Parametric Form?
Common Parametric Forms
| Curve | Parametric Form |
|---|---|
| Line segment | x = x₁ + (x₂−x₁)t, y = y₁ + (y₂−y₁)t, 0 ≤ t ≤ 1 |
| Circle | x = h + r cos t, y = k + r sin t, 0 ≤ t ≤ 2π |
| Ellipse | x = h + a cos t, y = k + b sin t |
| Parabola (horizontal) | x = at², y = 2at |
| Projectile motion | x = v₀ cos θ · t, y = v₀ sin θ · t − ½gt² |
Three Methods for Eliminating the Parameter
Method 1: Solve for t and substitute
x = 2t − 1, y = t² + 3
- 1.Solve the simpler equation for t: x = 2t − 1 → t = (x + 1)/2
- 2.Substitute into y = t² + 3:
- 3.y = ((x + 1)/2)² + 3
- 4.y = (x + 1)²/4 + 3
- 5.Answer: y = (x + 1)²/4 + 3 (a parabola, opening upward)
Restriction: t ∈ ℝ means x ∈ ℝ, so no domain restriction needed
Method 2: Pythagorean identity
x = 4 cos t, y = 3 sin t, 0 ≤ t ≤ 2π
- 1.Isolate the trig functions: cos t = x/4, sin t = y/3
- 2.Apply identity cos²t + sin²t = 1:
- 3.(x/4)² + (y/3)² = 1
- 4.Answer: x²/16 + y²/9 = 1 (an ellipse with a = 4, b = 3)
Full cycle: t from 0 to 2π traces the complete ellipse counterclockwise
Method 3: Trig identity (tan/sec)
x = sec t, y = tan t, −π/2 < t < π/2
- 1.Recall the identity: sec²t − tan²t = 1
- 2.Since x = sec t and y = tan t: x² − y² = 1
- 3.Answer: x² − y² = 1 (right branch of a hyperbola, since sec t ≥ 1 means x ≥ 1)
The domain restriction t ∈ (−π/2, π/2) limits the curve to the right branch
Orientation — Direction of Travel
Orientation is the direction the curve is traced as t increases. On graphs, orientation is shown with arrows pointing in the direction of increasing t.
Finding orientation
Plug in a few increasing values of t and plot. The sequence of points gives the direction.
Reversing orientation
Replace t with −t (or negating the parameter) reverses direction.
Key point: Two parametric sets can produce the same rectangular equation but different orientations. On exams, you must identify orientation from the parametric form — not from the rectangular form.
Projectile Motion — The Classic Application
Standard Projectile Equations
Worked Example
A ball is kicked at 60 ft/s at 30° above horizontal from ground level. Find: (a) max height, (b) time of flight, (c) range.
Setup: x = 60 cos(30°) · t = 30√3 · t, y = −16t² + 60 sin(30°) · t = −16t² + 30t
(a) Max height: dy/dt = −32t + 30 = 0 → t = 30/32 = 15/16 sec
y(15/16) = −16(15/16)² + 30(15/16) = −225/16 + 450/16 = 225/16 ≈ 14.1 ft
(b) Time of flight: y = 0 → −16t² + 30t = 0 → t(−16t + 30) = 0 → t = 0 or 15/8 sec
(c) Range: x(15/8) = 30√3 · (15/8) = 450√3/8 ≈ 97.4 ft
Frequently Asked Questions
What is the difference between parametric and rectangular equations?
A rectangular equation directly relates x and y, like y = x² + 1. A parametric equation expresses both x and y as separate functions of a third variable called the parameter (usually t): x = f(t), y = g(t). The parameter adds information that rectangular form loses — specifically, the direction of travel along the curve and time-dependent behavior. For example, x = cos t, y = sin t traces a unit circle counterclockwise, while x = cos(-t), y = sin(-t) traces the same circle clockwise. Rectangular form (x² + y² = 1) cannot show this distinction.
How do you eliminate the parameter from parametric equations?
There are three main methods: (1) Solve one equation for t, then substitute into the other — works well for linear x(t) or y(t). (2) Use a Pythagorean identity — works when x = a·cos t and y = b·sin t (use cos²t + sin²t = 1). (3) Use other trig identities — when equations involve tan, sec, etc. Example: x = 3cos t, y = 2sin t → x/3 = cos t, y/2 = sin t → (x/3)² + (y/2)² = 1, which is an ellipse.
What is orientation in a parametric curve?
Orientation is the direction in which the curve is traced as the parameter t increases. When you graph a parametric curve, you draw arrows showing this direction of travel. For example, x = t, y = t² has orientation moving left-to-right (x increases as t increases). Orientation is lost when you convert to rectangular form — y = x² looks the same whether you move left or right. On exams, orientation is shown with arrows on the curve.
Practice parametric equations
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