8.7Graphing Parametric Curves
Graph parametric curves including cycloids and Lissajous figures. Solve applied problems involving circular motion, satellite orbits, and other real-world parametric models.
Parametric curves describe complex paths that cannot be expressed as y = f(x). Cycloids, Lissajous figures, and epicycloids appear in physics, engineering, and computer graphics.
Essential Question
How do you graph a parametric curve by plotting points and tracking direction, and how do parametric equations model real-world motion like projectile trajectories?
Lesson Overview
Graphing Parametric Curves
- Make a table of (t, x, y) values by substituting key t-values.
- Plot the (x, y) points in the coordinate plane.
- Connect the points with arrows showing the direction of travel as t increases.
Key features to identify:
- Starting point (smallest t) and ending point (largest t)
- Direction of travel (orientation)
- Self-intersections — points where the curve crosses itself
Projectile Motion Model (no air resistance)
g ≈ 9.8 m/s² (metric) or 32 ft/s² (imperial)
Other Applications
- Circular motion: x = r cos(t), y = r sin(t)
- Lissajous figures: x = A sin(at + δ), y = B sin(bt) — figure-8 and more complex shapes
Worked Examples
Graph x = t², y = t − 1 for −3 ≤ t ≤ 3. Identify direction and any special points.
t = −3: (9, −4) t = −2: (4, −3) t = −1: (1, −2)
t = 0: (0, −1) t = 1: (1, 0) t = 2: (4, 1) t = 3: (9, 2)
Plot points and connect with arrows — curve moves upward as t increases.
Vertex (leftmost point) at (0, −1) when t = 0.
A ball is kicked at 20 m/s at 30° above horizontal. Find max height, range, and time of flight.
v₀ = 20, α = 30°; x(t) = 20 cos30° · t = 10√3 · t
y(t) = 20 sin30° · t − 4.9t² = 10t − 4.9t²
Max height at t = 10/9.8 ≈ 1.02 s: y ≈ 10(1.02) − 4.9(1.02)² ≈ 10.2 − 5.1 ≈ 5.1 m
Time of flight: 10t − 4.9t² = 0 → t(10 − 4.9t) = 0 → t = 20/9.8 ≈ 2.04 s
Range = 10√3 · 2.04 ≈ 17.32 · 2.04 ≈ 35.3 m
Graph x = 2cos(t), y = 3sin(2t) for t ∈ [0, 2π] (Lissajous figure). Identify key points.
t = 0: (2, 0) t = π/4: (√2, 3) t = π/2: (0, 0)
t = 3π/4: (−√2, 3) t = π: (−2, 0) t = 3π/2: (0, 0)
Curve passes through origin twice — figure-8 shape.
x ranges from −2 to 2; y ranges from −3 to 3.
A projectile is launched from a height of 50 m at 15 m/s horizontally. Find when it hits the ground and how far.
x(t) = 15t (horizontal, constant velocity)
y(t) = 50 − 4.9t² (vertical, launched horizontally so v_y = 0)
Set y = 0: 50 − 4.9t² = 0 → t² = 50/4.9 ≈ 10.2 → t ≈ 3.19 s
x = 15 · 3.19 ≈ 47.9 m
Graph x = sin(t), y = sin(2t) for t ∈ [0, 2π]. Find all self-intersections.
Self-intersection: find t₁ ≠ t₂ with sin(t₁) = sin(t₂) and sin(2t₁) = sin(2t₂).
At t = 0 and t = π: x = 0, y = 0 — same point (0, 0). ✓
At t = π/2: (1, 0); at t = 3π/2: (−1, 0) — different points, not intersections.
No other repeated (x, y) pairs exist on [0, 2π].
Guided Practice
Graph x = t + 1, y = t² − 4 for −3 ≤ t ≤ 3. Find the vertex.
Hint: Vertex occurs at the minimum y-value; find t where dy/dt = 0.
A ball is thrown at 25 m/s at 45°. Find the maximum height and range.
Hint: Use x(t) = v₀cosα · t and y(t) = v₀sinα · t − 4.9t².
Graph x = 3cos(t), y = 3sin(t) for t ∈ [0, π]. Describe the curve and direction.
Hint: This is a semicircle — which half? Check the starting and ending points.
For x = t², y = t³, find the point where the curve crosses itself (if any) and describe the behavior near t = 0.
Hint: Does the curve return to any point it has already visited? Check if two different t-values give the same (x, y).
A projectile is launched at 40 m/s at 60°. Find the time when it reaches its maximum height.
Hint: Maximum height occurs when the vertical velocity component equals zero.
Key Vocabulary
Parametric curve
The set of points (x(t), y(t)) traced as t varies over an interval.
Direction of travel
The direction in which the curve is traced as t increases; shown with arrows on the graph.
Orientation
The direction of travel along a parametric curve — clockwise or counterclockwise.
Projectile motion
Motion under gravity with x(t) = v₀cosα · t, y(t) = v₀sinα · t − ½gt² (no air resistance).
Lissajous figure
A parametric curve of the form x = A sin(at + δ), y = B sin(bt); produces figure-8 and more complex shapes.
Self-intersection
A point where the curve crosses itself — two different t-values give the same (x, y) point.
Range (projectile)
The horizontal distance traveled when the projectile returns to its launch height; R = v₀² sin(2α) / g.
Quick Check
Interactive Practice — 5 Questions
For x = cos(t), y = sin(t), t ∈ [0, 2π], the curve is traced:
A projectile launched at v₀ = 30 m/s, α = 90° has range:
Maximum range for a projectile (flat ground) occurs at launch angle:
The curve x = t², y = t⁴ is the same as the rectangular curve:
For projectile motion, the time of flight (launched and landing at same height) is:
Common Mistakes
Plotting (t, x) or (t, y) instead of (x, y)
Parametric curves are plotted in the xy-plane. Use t only to generate the (x, y) pairs — t itself is not an axis.
Forgetting to use g = 9.8 m/s² (or 32 ft/s²) consistently
Use g = 9.8 m/s² for metric units and g = 32 ft/s² for imperial. Mixing units gives wrong answers.
Assuming the curve is the same as the rectangular equation with no restrictions
Parametric equations may trace only part of the rectangular curve. Always check the t-interval.
Thinking maximum range occurs at 90°
Maximum range on flat ground occurs at 45°. At 90° the projectile goes straight up and has zero range.
Math Tips
To graph a parametric curve efficiently: find key t-values (start, end, where x or y is 0 or extremal), compute (x, y) at each, then connect with direction arrows.
For projectile motion, the horizontal and vertical components are independent. Horizontal: constant velocity. Vertical: constant acceleration due to gravity.
The range formula R = v₀²sin(2α)/g shows that complementary angles (e.g., 30° and 60°) give the same range.
Self-intersections occur when two different t-values give the same (x, y). Set up the system x(t₁) = x(t₂) and y(t₁) = y(t₂) and solve.
Lissajous figures arise when the frequency ratio a/b is rational. The number of loops relates to the ratio of frequencies.