๐Ÿ  Homeโ€บ Unit 1โ€บ 1.2 Notes
๐Ÿ“š Topic 1.2 ยท 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.

๐Ÿ“‹ Notes ๐Ÿƒ Flashcards ๐Ÿง  Practice Quiz
๐Ÿ“
Rate of Change at a Point
Positive ยท Zero ยท Negative โ€” the roller coaster model
๐ŸŽข

Think of the graph as a roller coaster

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

โฌ†๏ธ
Positive ROC
Function is increasing at that point. Going uphill on the roller coaster.
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Zero ROC
At a local max or min. The very peak or valley โ€” transitioning between going up and down.
โฌ‡๏ธ
Negative ROC
Function is decreasing at that point. Going downhill on the roller coaster.
๐Ÿ“

AP Precalculus vs. Calculus

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

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Ordering Rates of Change โ€” Example 1
Graph of f with three labeled points A, B, C
x y โˆ’3โˆ’2โˆ’1 12345 321 โˆ’1โˆ’2โˆ’3 A B C
Graph of f โ€” three labeled points
๐Ÿ“Œ Example 1 โ€” Order the rates of change at A, B, C from least to greatest.
A
Point A is on the rising (uphill) portion of the curve โ†’ rate of change is positive.
B
Point B is at the local maximum (top of the peak) โ†’ rate of change is zero. (Transitioning from going up to going down.)
C
Point C is on the falling (downhill) portion โ†’ rate of change is negative.
โ†’
From least to greatest: most negative first โ†’ C (negative) < B (zero) < A (positive).
B_rate < C_rate < A_rate  โ†’  Negative < Zero < Positive
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Comparing Steepness โ€” Example 2
Graph of g with four labeled points โ€” finding the least (most negative) ROC
๐ŸŽฏ

Key insight: "Least" means most negative

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

x y โˆ’4โˆ’3โˆ’2โˆ’1 12345 4321 โˆ’1โˆ’2โˆ’3 A B C D
Graph of g โ€” four labeled points
๐Ÿ“Œ Example 2 โ€” Of points A, B, C, D, at which is the rate of change of g the least?
A
Rising section โ€” ROC is positive. Can't be the least.
B
Still rising, but more gently than A โ€” ROC is positive but smaller. Still not least.
C
Near the local maximum โ€” ROC is close to zero (or small negative). Not the least.
D
On a steep downhill section โ€” ROC is negative and steep โ†’ this is the most negative = least.
Answer: (D) โ€” steepest downhill = most negative = least rate of change
๐Ÿ“Š
Average Rate of Change (AROC)
Over an interval โ€” not at a single point
AROC = ฮ”y / ฮ”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) โˆ’ f(a)) / (b โˆ’ a)
Formula version โ€” 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.
๐Ÿ’ก

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).

ใ€ฐ๏ธ
Greatest AROC from a Graph โ€” Example 3
Graph of h โ€” comparing four intervals
๐Ÿ“Œ Example 3 โ€” Graph of h on [0, 9]. Of the four intervals below, which has the greatest average rate of change?
A
2 โ‰ค x โ‰ค 3: h starts and ends at roughly the same value โ†’ outputs don't change โ†’ AROC โ‰ˆ zero.
B
3 โ‰ค x โ‰ค 5: h decreases (right endpoint is lower than left) โ†’ AROC is negative.
C
5 โ‰ค x โ‰ค 7: h increases significantly โ†’ AROC is positive. This is the only positive one โ€” it must be greatest.
D
7 โ‰ค x โ‰ค 8: h drops sharply โ†’ AROC is negative.
Answer: (C) 5 โ‰ค x โ‰ค 7 โ€” positive AROC, and the only positive interval = greatest
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Real-World & Analytical Examples
Examples 4, 5, 6
๐Ÿ“Œ Example 4 โ€” A function always has a negative rate of change. Which real-world scenario matches?
(A) child age โ†’ height   (B) points scored โ†’ time remaining   (C) time โ†’ height of ball   (D) radius โ†’ area
A
As a child gets older, height generally increases โ†’ ROC positive. โœ—
B
As more points are scored, the clock time decreases (time remaining drops) โ†’ ROC always negative. โœ“
C
Ball goes up then comes down โ†’ ROC is sometimes positive, sometimes negative. โœ—
D
As radius increases, area increases โ†’ ROC positive. โœ—
Answer: (B) โ€” more points scored โ†’ time remaining always decreasing โ†’ always negative ROC
๐Ÿ“Œ Example 5 โ€” k(x) = 3.16 + 4.2x โˆ’ 0.85xยฒ on [โˆ’10, 10]. Which statement is correct?
k(x) = 3.16 + 4.2x โˆ’ 0.85xยฒ
1
This is a downward-opening parabola (coefficient of xยฒ is โˆ’0.85 < 0). It has a maximum vertex.
2
Vertex (maximum) at x = โˆ’b/(2a) = โˆ’4.2/(2ยท(โˆ’0.85)) = โˆ’4.2/โˆ’1.7 โ‰ˆ 2.470.
3
Before the vertex (x < 2.470): function is increasing โ†’ positive ROC.
After the vertex (x > 2.470): function is decreasing โ†’ negative ROC.
D
The interval 2.470 โ‰ค x โ‰ค 10 is entirely to the right of the vertex โ†’ k is decreasing โ†’ negative ROC. โœ“
Answer: (D) k has a negative rate of change on 2.470 โ‰ค x โ‰ค 10
๐Ÿ“Œ Example 6 โ€” US female life expectancy over 50-year intervals. On which interval is the AROC greatest?
BirthyearLife ExpectancyAROC = ฮ”y / 50
1800 โ†’ 185041.24 โ†’ 46.10(46.10 โˆ’ 41.24) / 50 = 0.0972
1850 โ†’ 190046.10 โ†’ 53.63(53.63 โˆ’ 46.10) / 50 = 0.1506
1900 โ†’ 195053.63 โ†’ 70.65(70.65 โˆ’ 53.63) / 50 = 0.3404 โ† Greatest
1950 โ†’ 200070.65 โ†’ 81.83(81.83 โˆ’ 70.65) / 50 = 0.2236
Answer: (C) 1900 to 1950 โ€” AROC = 0.3404 years per year (fastest increase in life expectancy)
โšก
Quick Reference
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๐Ÿ”‘ Key Rules
Going up โ†’ positive ROC
Going down โ†’ negative ROC
Peak or valley โ†’ ROC = 0
AROC = (f(b) โˆ’ f(a)) / (b โˆ’ a)
"Least" ROC = most negative
โš ๏ธ Common Mistakes
โŒ "Least ROC means smallest positive"
"Least" means most negative on the number line. A steep downhill (โˆ’5) is less than a gentle downhill (โˆ’1).
โŒ ROC = 0 only when function = 0
ROC = 0 at LOCAL EXTREMA (peaks and valleys), regardless of the y-value at that point.
โŒ AROC positive = always increasing
AROC can be positive even if the function dips in the middle โ€” only the START and END values matter for AROC.
โŒ 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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