🏠 Homeβ€Ί Unit 1β€Ί 1.2 Notes
πŸ“š Topic 1.2 Β· Unit 1

Rates of
Change

Rate of change at a point (positive/negative/zero), comparing ROC at two points, and average rate of change over an interval.

πŸ“‹ Notes πŸƒ Flashcards 🧠 Practice Quiz
🏺
Change in Tandem β€” Example 1
Water fills a vase at a constant rate β€” which graph matches?
πŸ’‘

When water fills at a constant rate, the height depends on the width of the vase at that level. Narrow section β†’ height rises quickly. Wide section β†’ height rises slowly.

πŸ“Œ 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?
wide β†’ slow rise narrow β†’ fast rise Vase (side view)
(A)
t h
(B)
t h
(C)
t h
(D) βœ“
t h
Key
Water fills at a constant rate. The width of the vase at each level determines how fast the height rises β€” narrow = fast rise, wide = slow rise.
A βœ—
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. βœ—
B βœ—
Concave up then concave down: height rises faster first, then slower β€” but the wide base is at the bottom (slow rise should come first, not second). Backwards. βœ—
C βœ—
S-shape with flat middle: implies height barely changes at some point β€” impossible if water is filling at a constant rate. βœ—
D βœ“
Concave down β†’ concave up: starts slow (wide base β†’ height rises slowly β†’ concave down ∩), then rises faster (narrow neck β†’ height rises quickly β†’ concave up βˆͺ). βœ“
Answer: (D) β€” height increases at a decreasing rate (wide base), then at an increasing rate (narrow neck)
πŸ“
Rate of Change at a Point
Positive Β· Zero Β· Negative β€” the roller coaster model
🎒

Think of the graph as a roller coaster

Going up at that point β†’ rate of change is positive.
Going down at that point β†’ rate of change is negative.
At the very top of a peak or bottom of a valley β†’ you are not moving up or down at that exact instant β†’ rate of change is zero.

⬆️
Positive ROC
Function is increasing at that point. Going uphill on the roller coaster.
➑️
Zero ROC
At a local max or min. The very peak or valley β€” transitioning between going up and down.
⬇️
Negative ROC
Function is decreasing at that point. Going downhill on the roller coaster.
πŸ“

AP Precalculus vs. Calculus

In AP Precalculus, we cannot compute the exact rate of change at a single point β€” 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.

πŸ“ˆ
Ordering Rates of Change β€” Example 1
Graph of f with three labeled points A, B, C
x y βˆ’3βˆ’2βˆ’1 012 321 βˆ’1βˆ’2 A rising β†’ positive B peak β†’ slope = 0 C falling β†’ negative Least to greatest: C < B < A
Graph of f β€” A rising (positive ROC), B at peak (zero ROC), C falling (negative ROC)
πŸ“Œ Example 1 β€” 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 β†’ the tangent line slopes upward β†’ rate of change is positive.
B
Point B is at the local maximum β€” the very top of the peak. The curve transitions from going up to going down, so the tangent is flat β†’ rate of change is zero.
C
Point C is on the downhill (falling) portion β†’ the tangent line slopes downward β†’ rate of change is negative.
β†’
Ordering from least (most negative) to greatest: C (negative) < B (zero) < A (positive).
Least to greatest: C < B < A  β†’  negative < zero < positive
πŸ”
Comparing Steepness β€” Example 2
Graph of g with four labeled points β€” finding the least (most negative) ROC
🎯

Key insight: "Least" means most negative

When comparing rates of change, least means the most negative value β€” the steepest downhill slope. A very steep downhill is a smaller (more negative) number than a gentle downhill.

x y βˆ’4βˆ’3βˆ’2 βˆ’101 234 5 321 βˆ’1βˆ’2 A steep rise B gentle rise C near peak β‰ˆ 0 D steep fall Least ROC = D (steepest downhill = most negative)
Graph of g β€” A steep rise, B gentle rise, C near peak, D steep fall
πŸ“Œ Example 2 β€” Of points A, B, C, D, at which is the rate of change of g the least?
A
Rising section β€” ROC is positive. Can't be the least.
B
Still rising, but more gently than A β€” ROC is positive but smaller. Still not least.
C
Near the local maximum β€” ROC is close to zero (or small negative). Not the least.
D
On a steep downhill section β€” ROC is negative and steep β†’ this is the most negative = least.
Answer: (D) β€” steepest downhill = most negative = least rate of change
πŸ“Š
Average Rate of Change (AROC)
Over an interval β€” not at a single point
AROC = Ξ”y / Ξ”x
The constant rate of change that produces the same total change in output as the function did over that interval. It's the slope of the secant line connecting the two endpoints.
AROC = (f(b) βˆ’ f(a)) / (b βˆ’ a)
Formula version β€” output change divided by input change. In Topic 1.2 we only determine if AROC is positive, negative, or zero, and compare two intervals. Full calculation comes in Topic 1.3.
πŸ’‘

How to tell if AROC is positive/negative/zero

AROC positive: output at the right endpoint is greater than at the left endpoint (overall going up on the interval).
AROC negative: output at the right endpoint is less (overall going down).
AROC zero: outputs at both endpoints are equal (same start and end, even if it wiggled in between).

〰️
Greatest AROC from a Graph β€” Example 3
Graph of h β€” comparing four intervals
123 456 789 x 321 βˆ’1βˆ’2βˆ’3 y A: β‰ˆ0 B: neg C: pos βœ“ D: neg
Graph of h on [0, 9] β€” shaded regions A–D with secant lines showing AROC direction
πŸ“Œ Example 3 β€” Graph of h on [0, 9]. Of the four intervals below, which has the greatest average rate of change?
A
2 ≀ x ≀ 3: h starts and ends at roughly the same value β†’ outputs don't change β†’ AROC β‰ˆ zero.
B
3 ≀ x ≀ 5: h decreases (right endpoint is lower than left) β†’ AROC is negative.
C
5 ≀ x ≀ 7: h increases significantly β†’ AROC is positive. This is the only positive one β€” it must be greatest.
D
7 ≀ x ≀ 8: h drops sharply β†’ AROC is negative.
Answer: (C) 5 ≀ x ≀ 7 β€” positive AROC, and the only positive interval = greatest
πŸ“Š
Full Behavior Analysis β€” Example 4
Graph of p on [0, 10] Β· Label each section: sign, direction, concavity, ROC
🎯

The four questions to ask about each section

1. Is the function positive or negative? (above or below the x-axis)
2. Is the function increasing or decreasing? (going uphill or downhill)
3. Is the rate of change increasing or decreasing? (concave up βˆͺ or concave down ∩)
4. Therefore: is the graph concave up or concave down?

x y 012 345 678 910 21 βˆ’1βˆ’2 A B C D E (0,βˆ’1) (2, 2) (4, 0) (6,βˆ’2) (8, 0) (10, 2)
Graph of p on [0, 10] β€” five labeled sections A through E
Section Interval p positive or negative? p increasing or decreasing? ROC increasing or decreasing? Concave up or down?
A 0 < x < 2 Negative (x=0 to ~x=0.9)
then Positive (~x=0.9 to x=2)		 Increasing ↑
(outputs going up)		 Increasing
(slope getting steeper)		 Concave Up βˆͺ
B 2 < x < 4 Positive
(graph above x-axis)		 Decreasing ↓
(outputs going down)		 Decreasing
(slope getting less steep)		 Concave Down ∩
C 4 < x < 6 Negative
(graph below x-axis)		 Decreasing ↓
(outputs going down)		 Increasing
(slope becoming less negative)		 Concave Up βˆͺ
D 6 < x < 8 Negative
(graph below x-axis)		 Increasing ↑
(outputs going up)		 Increasing
(slope getting steeper)		 Concave Up βˆͺ
E 8 < x < 10 Positive
(graph above x-axis)		 Increasing ↑
(outputs going up)		 Decreasing
(slope getting less steep)		 Concave Down ∩
πŸ’‘

Key takeaway from section C

Section C is negative AND decreasing AND concave up. This is a common AP trap β€” the function is falling AND the rate of change is increasing at the same time. "Decreasing" refers to the outputs going down. "ROC increasing" means the slope is becoming less negative (e.g. going from βˆ’3 to βˆ’1). These are independent facts that can combine in any way.

πŸ“ˆ
Graph of k β€” Examples 5–8
k on [0,9] Β· Points A(1,βˆ’3) B(2,0) C(5,3) D(7,βˆ’1) E(8,βˆ’3) F(9,βˆ’1)
123 456 789 x 321 βˆ’1βˆ’2βˆ’3 y A B C D E F 0
Graph of k on [0, 9] β€” A(1,βˆ’3) B(2,0) C(5,3) D(7,βˆ’1) E(8,βˆ’3) F(9,βˆ’1)
πŸ“Œ Example 5 β€” On which interval is k negative and decreasing?
(A) A to B   (B) B to C   (C) D to E   (D) E to F
A→B
A(1,βˆ’3)β†’B(2,0): going from βˆ’3 up to 0. Negative (below x-axis) βœ“, but increasing. βœ—
B→C
B(2,0)β†’C(5,3): going up, positive values. Neither negative nor decreasing. βœ—
D→E
D(7,βˆ’1)β†’E(8,βˆ’3): going from βˆ’1 down to βˆ’3. Both values negative βœ“, going downhill βœ“. Answer: D to E.
E→F
E(8,βˆ’3)β†’F(9,βˆ’1): going up (from βˆ’3 to βˆ’1). Negative βœ“, but increasing. βœ—
Answer: (C) D to E β€” k is negative and decreasing
πŸ“Œ Example 6 β€” Which statement about the rate of change of k is true?
(A) Rate of change of k is negative on A to B   (B) Rate of change of k is negative on B to C   (C) Rate of change of k is positive on D to E   (D) Rate of change of k is positive on E to F
A
A(1,βˆ’3)β†’B(2,0): going uphill β†’ ROC is positive. Option says negative. βœ—
B
B(2,0)β†’C(5,3): going uphill β†’ ROC is positive. Option says negative. βœ—
C
D(7,βˆ’1)β†’E(8,βˆ’3): going downhill β†’ ROC is negative. Option says positive. βœ—
D
E(8,βˆ’3)β†’F(9,βˆ’1): going from βˆ’3 up to βˆ’1 β†’ uphill β†’ ROC is positive. βœ“
Answer: (D) β€” rate of change of k is positive on E to F
πŸ“Œ Example 7 β€” Which statement about the rate of change of k is true?
(A) Rate of change is decreasing on A to B   (B) Rate of change is increasing on B to C   (C) Rate of change is increasing on D to E   (D) Rate of change is decreasing on E to F
Key
ROC increasing = concave up βˆͺ. ROC decreasing = concave down ∩.
A
Aβ†’B: curve bends concave up βˆͺ β†’ ROC increasing. Option says decreasing. βœ—
B
Bβ†’C: curve arches over ∩ (concave down) β†’ ROC decreasing. Option says increasing. βœ—
C
Dβ†’E: k falls but curve bends upward βˆͺ (concave up) β†’ ROC is increasing (slope becoming less negative). βœ“
D
Eβ†’F: curve bends concave up βˆͺ β†’ ROC increasing. Option says decreasing. βœ—
Answer: (C) rate of change of k is increasing on D to E
πŸ“Œ Example 8 β€” On which interval is k increasing AND the graph of k concave down?
(A) A to B   (B) B to C   (C) C to D   (D) D to E
A→B
Going uphill βœ“ (increasing). Curve bends like a βˆͺ (concave up). βœ—
B→C
Going uphill βœ“ (increasing). Curve arches over like a hill ∩ (concave down) βœ“. Answer: B to C.
Answer: (B) k is increasing and concave down on B to C
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Real-World & Analytical Examples
Examples 4, 5, 6
πŸ“Œ Example 4 β€” A function always has a negative rate of change. Which real-world scenario matches?
(A) child age β†’ height   (B) points scored β†’ time remaining   (C) time β†’ height of ball   (D) radius β†’ area
A
As a child gets older, height generally increases β†’ ROC positive. βœ—
B
As more points are scored, the clock time decreases (time remaining drops) β†’ ROC always negative. βœ“
C
Ball goes up then comes down β†’ ROC is sometimes positive, sometimes negative. βœ—
D
As radius increases, area increases β†’ ROC positive. βœ—
Answer: (B) β€” more points scored β†’ time remaining always decreasing β†’ always negative ROC
πŸ“Œ Example 5 β€” k(x) = 3.16 + 4.2x βˆ’ 0.85xΒ² on [βˆ’10, 10]. Which statement is correct?
(A) k has a positive rate of change on [βˆ’10, 10]   (B) k has a negative rate of change on [βˆ’10, 2.470]   (C) k has a positive rate of change on [2.470, 10]   (D) k has a negative rate of change on [2.470, 10]
k(x) = 3.16 + 4.2x βˆ’ 0.85xΒ²
1
This is a downward-opening parabola (coefficient of xΒ² is βˆ’0.85 < 0) β€” it has one maximum and no minimum.
2
Vertex (maximum) at x = βˆ’b/(2a) = βˆ’4.2/(2Β·(βˆ’0.85)) β‰ˆ 2.470.
3
Left of vertex (x < 2.470): k is increasing β†’ positive ROC. Right of vertex (x > 2.470): k is decreasing β†’ negative ROC.
D βœ“
[2.470, 10] is entirely to the right of the vertex β†’ k is decreasing throughout β†’ negative ROC. βœ“
Answer: (D) k has a negative rate of change on 2.470 ≀ x ≀ 10
πŸ“Œ Example 6 β€” US female life expectancy. On which 50-year interval is the average rate of change the greatest?
(A) 1800–1850   (B) 1850–1900   (C) 1900–1950   (D) 1950–2000
OptionIntervalLife ExpectancyAROC = Ξ”y / 50
(A)1800 β†’ 185041.24 β†’ 46.10(46.10 βˆ’ 41.24) / 50 = 0.097
(B)1850 β†’ 190046.10 β†’ 53.63(53.63 βˆ’ 46.10) / 50 = 0.151
(C) βœ“1900 β†’ 195053.63 β†’ 70.65(70.65 βˆ’ 53.63) / 50 = 0.340 ← Greatest
(D)1950 β†’ 200070.65 β†’ 81.83(81.83 βˆ’ 70.65) / 50 = 0.224
Answer: (C) 1900–1950 β€” AROC = 0.340 years/year, the largest of all four intervals
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Quick Reference
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πŸ”‘ Key Rules
Going up β†’ positive ROC
Going down β†’ negative ROC
Peak or valley β†’ ROC = 0
AROC = (f(b) βˆ’ f(a)) / (b βˆ’ a)
"Least" ROC = most negative
⚠️ Common Mistakes
❌ "Least ROC means smallest positive"
"Least" means most negative on the number line. A steep downhill (βˆ’5) is less than a gentle downhill (βˆ’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 β€” 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).
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