AP PRECALCULUS — UNIT 4A · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.3 — Parametric Functions and Rates of Change
Notes — How Position Changes with Time
💡 Learning Objectives (4.3.A)
By the end of this lesson you will be able to:
- Compute the average rate of change of x(t) and of y(t) over a t-interval
- Interpret these rates as horizontal and vertical components of motion
- Compute the average rate of change of y with respect to x along the path
- Use rates of change to describe a particle's motion qualitatively
1. Rates of Change of Each Coordinate
For a parametric function (x(t), y(t)), the average rate of change of each coordinate over [t₁, t₂] is exactly the average rate of change of an ordinary function:
AROC of x: (x(t₂) − x(t₁)) / (t₂ − t₁)
AROC of y: (y(t₂) − y(t₁)) / (t₂ − t₁)
These two numbers describe how rapidly the particle is moving HORIZONTALLY and VERTICALLY on average over the interval. Their signs indicate direction (positive = right or up, negative = left or down).
2. Worked Example
📘 Example — Compute AROC over [1, 4]
A particle has (x(t), y(t)) = (t² + 1, 2t − 5).
- x(1) = 2, x(4) = 17. AROC of x = (17 − 2)/(4 − 1) = 5
- y(1) = −3, y(4) = 3. AROC of y = (3 − (−3))/(4 − 1) = 2
- Interpretation: on average, the particle moves right by 5 units per unit time and up by 2 units per unit time over this interval.
3. Rate of Change of y with Respect to x
Often we want to know how y changes per unit change in x along the path — this is the average SLOPE of the curve over the interval:
AROC of y w.r.t. x: (y(t₂) − y(t₁)) / (x(t₂) − x(t₁))
Note: this is the same as (AROC of y) ÷ (AROC of x), provided the AROC of x is nonzero.
📘 Example — Slope over an interval
Using the previous (x(t), y(t)) = (t² + 1, 2t − 5):
- AROC of y w.r.t. x = (3 − (−3))/(17 − 2) = 6/15 = 2/5
- So on average over [1, 4], the path's slope is 2/5.
4. Direction of Motion
- If both AROCs are positive: motion is to the upper right
- If AROC of x positive, AROC of y negative: motion is to the lower right
- If AROC of x negative, AROC of y positive: motion is to the upper left
- If both AROCs are negative: motion is to the lower left
- If AROC of x is zero: motion is purely vertical over that interval
- If AROC of y is zero: motion is purely horizontal over that interval
5. When the Coordinate Functions Are Not Monotonic
If x(t) increases on part of [t₁, t₂] and decreases on the rest, the AVERAGE RATE OF CHANGE of x can still be computed — but it doesn't tell you direction at every moment, only the net effect over the whole interval.
📘 Example — Mixed-direction motion
A particle has x(t) = t² for t ∈ [−1, 2].
- x(−1) = 1, x(2) = 4. AROC = (4 − 1)/(2 − (−1)) = 1
- But x decreases from 1 to 0 on [−1, 0] and increases from 0 to 4 on [0, 2]
- Net AROC of 1 reflects the OVERALL change, not the moment-to-moment motion.
6. Geometric Meaning of AROCs
Picture the parametric curve in the plane. Connect the points P(t₁) = (x(t₁), y(t₁)) and P(t₂) = (x(t₂), y(t₂)) with a straight line. The slope of that secant line is exactly:
- AROC of y w.r.t. x = Δy / Δx
So this rate measures the slope of the chord from one position to another along the path.
7. Compare to AP-Style Questions
Typical AP questions on this topic:
- ‘Compute the average rate of change of x (or y) over [a, b].’
- ‘During the interval [a, b], on average is the particle moving up or down? Left or right?’
- ‘Find the average slope of the path over [a, b].’
- ‘At what value of t in [a, b] does the particle stop moving horizontally? Vertically?’ (Find where x'(t) = 0 conceptually — for AP precalc this is identified from a graph or the algebraic structure.)
8. Summary
- AROC of x over [t₁, t₂] = (x(t₂) − x(t₁))/(t₂ − t₁); same formula for y
- AROC of y w.r.t. x = Δy/Δx — the slope of the chord between two positions on the path
- Sign combinations of the AROCs determine the rough direction of motion (NE, SE, NW, SW)
- Average rates summarize NET behavior over an interval, even if motion is non-monotonic within