Change in Tandem — Example 1
Water fills a vase at a constant rate — which graph matches?
When water fills at a constant rate, the height depends on the width of the vase at that level. Narrow section → height rises quickly. Wide section → height rises slowly.
📌 The vase below is narrow at the top and wide at the bottom. Water fills at a constant rate. Which graph (A–D) correctly shows height vs. time?
(A)
(B)
(C)
(D) ✓
Key
Water fills at a constant rate. The width of the vase at each level determines how fast the height rises — narrow = fast rise, wide = slow rise.A ✗
Linear (straight line): implies a constant rate of change of height throughout. But the vase changes width, so the rate of height change must change. ✗B ✗
Concave up then concave down: height rises faster first, then slower — but the wide base is at the bottom (slow rise should come first, not second). Backwards. ✗C ✗
S-shape with flat middle: implies height barely changes at some point — impossible if water is filling at a constant rate. ✗D ✓
Concave down → concave up: starts slow (wide base → height rises slowly → concave down ∩), then rises faster (narrow neck → height rises quickly → concave up ∪). ✓Answer: (D) — height increases at a decreasing rate (wide base), then at an increasing rate (narrow neck)
Rate of Change at a Point
Positive · Zero · Negative — the roller coaster model
Think of the graph as a roller coaster
Going up at that point → rate of change is positive.
Going down at that point → rate of change is negative.
At the very top of a peak or bottom of a valley → you are not moving up or down at that exact instant → rate of change is zero.
Positive ROC
Function is increasing at that point. Going uphill on the roller coaster.
Zero ROC
At a local max or min. The very peak or valley — transitioning between going up and down.
Negative ROC
Function is decreasing at that point. Going downhill on the roller coaster.
AP Precalculus vs. Calculus
In AP Precalculus, we cannot compute the exact rate of change at a single point — that requires calculus (it's called the derivative or instantaneous rate of change). We can only say whether it is positive, negative, or zero, and compare rates at two different points.
Ordering Rates of Change — Example 1
Graph of f with three labeled points A, B, C
Graph of f — A rising (positive ROC), B at peak (zero ROC), C falling (negative ROC)
📌 Example 1 — Order the rates of change at A, B, C from least to greatest.
A
Point A is on the uphill (rising) portion of the curve → the tangent line slopes upward → rate of change is positive.B
Point B is at the local maximum — the very top of the peak. The curve transitions from going up to going down, so the tangent is flat → rate of change is zero.C
Point C is on the downhill (falling) portion → the tangent line slopes downward → rate of change is negative.→
Ordering from least (most negative) to greatest: C (negative) < B (zero) < A (positive).Least to greatest: C < B < A → negative < zero < positive
Comparing Steepness — Example 2
Graph of g with four labeled points — finding the least (most negative) ROC
Key insight: "Least" means most negative
When comparing rates of change, least means the most negative value — the steepest downhill slope. A very steep downhill is a smaller (more negative) number than a gentle downhill.
Graph of g — A steep rise, B gentle rise, C near peak, D steep fall
📌 Example 2 — Of points A, B, C, D, at which is the rate of change of g the least?
A
Rising section — ROC is positive. Can't be the least.B
Still rising, but more gently than A — ROC is positive but smaller. Still not least.C
Near the local maximum — ROC is close to zero (or small negative). Not the least.D
On a steep downhill section — ROC is negative and steep → this is the most negative = least.Answer: (D) — steepest downhill = most negative = least rate of change
Quick Reference
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🔑 Key Rules
Going up → positive ROC
Going down → negative ROC
Peak or valley → ROC = 0
AROC = (f(b) − f(a)) / (b − a)
"Least" ROC = most negative
⚠️ Common Mistakes
❌ "Least ROC means smallest positive"
"Least" means most negative on the number line. A steep downhill (−5) is less than a gentle downhill (−1).
❌ ROC = 0 only when function = 0
ROC = 0 at LOCAL EXTREMA (peaks and valleys), regardless of the y-value at that point.
❌ AROC positive = always increasing
AROC can be positive even if the function dips in the middle — only the START and END values matter for AROC.
❌ Can find exact ROC at a point in Precalc
In AP Precalculus, we can only say positive/negative/zero and compare. The exact value requires calculus (derivatives).