Rucete ✏ AP Precalculus In a Nutshell
10. Parametric Functions
This chapter introduces parametric equations and how they model motion. You’ll learn to graph parametric curves, analyze particle paths, find intercepts and extrema, convert between forms, and describe conic sections parametrically.
- Parametric Functions and Graphing
- Parametric equations define x and y as separate functions of a third variable t (the parameter): f(t) = (x(t), y(t))
- Useful when standard functions cannot fully describe motion
- Example: x(t) = 3t − 4, y(t) = t² + 1
- Graphing steps:
- Create a value table of t
- Plot points in order
- Add arrows for direction of motion
- Domains may restrict t and define start/end points
- Motion Modeling and Extrema
- Position at time t = a → plug into x(t), y(t)
- Relative extrema from max/min of x(t), y(t)
- Intercepts:
- x(t) = 0 → y-intercept
- y(t) = 0 → x-intercept
- Direction:
- x(t) increasing → right; decreasing → left
- y(t) increasing → up; decreasing → down
- Average Rate of Change and Slope
- Average rate of change: (x₂ − x₁)/(t₂ − t₁), (y₂ − y₁)/(t₂ − t₁)
- Slope between t-values: Δy/Δx
- Secant line approximates curve between points
- Parametric Forms of Lines and Circles
- Line segment from A to B:
x(t) = x₁ + (x₂ − x₁)t, y(t) = y₁ + (y₂ − y₁)t, t ∈ [0, 1] - Circle centered at (h, k): x(t) = h + a cos t, y(t) = k + a sin t
- Unit circle: x = cos t, y = sin t
- Implicit Functions and Intercepts
- Implicit example: x² + y² = 1
- Graph by solving or plotting values
- Solve x(t) = 0 or y(t) = 0 for intercepts
- Converting Forms
- Parametric → Rectangular:
- Solve x(t) or y(t) for t
- Substitute into the other equation
- Rectangular → Parametric:
- Let x = t, solve y = f(t)
- Or express both in terms of t
- Used to identify shapes (lines, circles, ellipses)
- Conic Sections
- Parabola:
- Standard: y² = ax or x² = ay
- Parametric: x = t, y = at² or x = at², y = t
- Ellipse:
- Standard: (x − h)²/a² + (y − k)²/b² = 1
- Parametric: x = h + a cos t, y = k + b sin t
- Hyperbola:
- Standard: (x − h)²/a² − (y − k)²/b² = 1
- Parametric: x = h + a sec t, y = k + b tan t
- For vertical hyperbolas: switch sec and tan
- Inverses and Representations
- Function f(x) → (x(t), y(t)) = (t, f(t))
- Inverse f⁻¹(x) → (x(t), y(t)) = (f(t), t)
- Same curve can be defined with different parameterizations
- Direction of motion depends on parameter t
In a Nutshell
Parametric functions use a third variable to describe motion and curves that can’t always be expressed as y = f(x). Whether you're modeling particle movement, graphing conics, or converting between forms, understanding parametric equations enhances your ability to interpret dynamic systems and complex shapes algebraically and graphically.