8.6Understanding Parametric Equations
Graph parametric curves (x(t), y(t)), eliminate the parameter to find the rectangular equation, and model projectile motion with x = v₀cosθ·t and y = v₀sinθ·t − ½gt².
Parametric equations describe motion along a curve — position as a function of time. They are essential for modeling projectile motion, circular motion, and are the foundation for parametric calculus (arc length, velocity vectors).
Essential Question
How do parametric equations x = f(t) and y = g(t) describe a curve differently from a single equation in x and y, and what information do they capture that rectangular equations cannot?
Lesson Overview
Parametric equations define a curve by expressing both x and y as functions of a third variable called the parameter, usually t:
The parametric curve is the set of all points (f(t), g(t)) as t varies over its interval. Unlike y = f(x), parametric equations can represent curves that loop, cross themselves, or fail the vertical line test.
Direction of travel: as t increases, the curve is traced in a specific direction — this orientation is encoded in the parametric form but lost when converting to a rectangular equation.
Eliminating the parameter: to find the rectangular equation, solve for t in one equation and substitute into the other. For trig forms, use the identity sin²t + cos²t = 1.
| Curve | x(t) | y(t) | Rectangular |
|---|---|---|---|
| Line | x₀ + at | y₀ + bt | y − y₀ = (b/a)(x − x₀) |
| Circle | r cos(t) | r sin(t) | x² + y² = r² |
| Ellipse | a cos(t) | b sin(t) | x²/a² + y²/b² = 1 |
| Parabola | t | t² | y = x² |
| Parabola (horiz.) | t² | t | x = y² |
Restricting the domain: by limiting the t-interval, parametric equations can represent only a portion of a rectangular curve — for example, just the upper semicircle or a finite line segment.
Worked Examples
Sketch the curve x = t − 1, y = t² for −2 ≤ t ≤ 2. Eliminate the parameter.
Make a table: t = −2 → (−3, 4); t = −1 → (−2, 1); t = 0 → (−1, 0); t = 1 → (0, 1); t = 2 → (1, 4).
Plot the points and connect in order of increasing t — the curve moves right as t increases.
Eliminate: from x = t − 1, solve for t: t = x + 1.
Substitute into y = t²: y = (x + 1)².
Domain: t ∈ [−2, 2] → x ∈ [−3, 1], so the rectangular equation is restricted to −3 ≤ x ≤ 1.
Eliminate the parameter from x = 3cos(t), y = 2sin(t) and identify the curve.
Isolate the trig functions: cos(t) = x/3, sin(t) = y/2.
Use the Pythagorean identity: cos²(t) + sin²(t) = 1.
Substitute: (x/3)² + (y/2)² = 1.
Simplify: x²/9 + y²/4 = 1.
Write parametric equations for the line through (2, −1) with slope 3.
A line through (x₀, y₀) with slope m = b/a can be parameterized with direction vector ⟨1, m⟩.
Choose a = 1, b = 3 (slope = b/a = 3).
Set x = x₀ + at = 2 + t, y = y₀ + bt = −1 + 3t.
Eliminate the parameter from x = eᵗ, y = e²ᵗ − 1 and identify the curve.
From x = eᵗ, note that eᵗ = x, so e²ᵗ = (eᵗ)² = x².
Substitute into y: y = x² − 1.
Domain restriction: since x = eᵗ {'>'} 0 for all t, we have x {'>'} 0.
Write parametric equations for the circle x² + y² = 25, then for only the upper semicircle.
Full circle: use x = r cos(t), y = r sin(t) with r = 5.
Full circle: x = 5cos(t), y = 5sin(t), t ∈ [0, 2π].
Upper semicircle: y ≥ 0 means sin(t) ≥ 0, which holds for t ∈ [0, π].
Upper semicircle: x = 5cos(t), y = 5sin(t), t ∈ [0, π].
Guided Practice
Sketch x = 2t + 1, y = t − 3 for −2 ≤ t ≤ 2. Eliminate the parameter and identify the curve.
Hint: Solve for t from the x-equation: t = (x − 1)/2. Then substitute into y = t − 3.
Eliminate the parameter from x = cos(t), y = sin²(t) and write the rectangular equation.
Hint: Use the identity sin²t = 1 − cos²t, then replace cos(t) with x.
Write parametric equations for the ellipse x²/16 + y²/9 = 1.
Hint: Match to the standard form x = a cos(t), y = b sin(t). Identify a² = 16 and b² = 9.
Eliminate the parameter from x = t², y = t³ and identify any domain restrictions.
Hint: Solve for t from x = t²: t = ±√x. Then substitute into y = t³. Consider which branch applies.
Write parametric equations for the line segment from (1, 2) to (5, 8).
Hint: Use x = x₁ + (x₂ − x₁)t, y = y₁ + (y₂ − y₁)t with t ∈ [0, 1]. At t = 0 you get the start point; at t = 1 you get the end point.
Key Vocabulary
Parametric equations
A pair of equations x = f(t) and y = g(t) that express the coordinates of a point as functions of a third variable t.
Example: x = cos(t), y = sin(t)
Parameter
The independent variable t in parametric equations. It is not necessarily time, though it often represents time in physics applications.
Example: In x = 3t + 1, y = t², the parameter is t.
Parametric curve
The set of all points (f(t), g(t)) traced as t varies over its domain interval.
Direction of travel (orientation)
The direction in which the curve is traced as t increases. This information is lost when converting to a rectangular equation.
Example: x = cos(t), y = sin(t) traces counterclockwise; x = cos(t), y = −sin(t) traces clockwise.
Eliminating the parameter
The process of combining the two parametric equations to produce a single rectangular equation in x and y only.
Example: From x = t + 1, y = t²: t = x − 1, so y = (x − 1)².
Rectangular equation
An equation relating x and y directly, without a parameter. Also called a Cartesian equation.
Example: y = x² − 4x + 3
Check Your Understanding
Interactive Practice — 5 Questions
The curve defined by x = cos(t), y = sin(t) for t ∈ [0, 2π] is a:
Eliminating the parameter from x = t + 2, y = t² gives:
Parametric equations can represent curves that:
The parametric equations x = 4cos(t), y = 3sin(t) represent:
To eliminate the parameter from x = eᵗ, y = e³ᵗ, the best first step is:
Common Mistakes
Forgetting the domain restriction when eliminating the parameter — e.g., from x = t², y = t, writing y = √x for all x.
Since x = t² ≥ 0 and y = t can be negative, the correct rectangular form is y = ±√x (or two branches). If t ≥ 0, then y = √x; if t ≤ 0, then y = −√x. Always check the t-interval.
Assuming the parameter t always represents time.
t is just a variable — it can represent time, angle, arc length, or anything else. The label 't' is conventional, not mandatory.
Losing the direction of travel when converting to rectangular form.
The rectangular equation x²/9 + y²/4 = 1 gives the same ellipse whether traced clockwise or counterclockwise. Always note the orientation from the original parametric form.
For x = a cos(t), y = b sin(t), writing the ellipse as x²/b² + y²/a² = 1 (swapping a and b).
Divide correctly: cos(t) = x/a, sin(t) = y/b, so (x/a)² + (y/b)² = 1, giving x²/a² + y²/b² = 1.
Math Tips
The three main elimination strategies: (1) solve for t directly (linear/polynomial), (2) use sin²t + cos²t = 1 (trig forms), (3) use eᵗ substitution (exponential forms).
Parametric equations are more powerful than rectangular equations — they encode direction, speed, and can represent curves that fail the vertical line test.
For a line through (x₀, y₀) with direction vector ⟨a, b⟩: x = x₀ + at, y = y₀ + bt. The slope of the line is b/a (when a ≠ 0).
To restrict a curve to a portion, restrict the t-interval. The full circle x = cos(t), y = sin(t) uses t ∈ [0, 2π]; the upper semicircle uses t ∈ [0, π]; the right semicircle uses t ∈ [−π/2, π/2].
When eliminating the parameter using trig, always look for sin²t + cos²t = 1 — this is the key identity for circles and ellipses. For hyperbolas, use sec²t − tan²t = 1.