AP Precalculus 1.3 Rates of Change in Linear Functions — Notes

🔍

Linear · Quadratic · Neither

What the AROC pattern tells you about the function type

How to Classify a Function Using AROC Over Equal-Length Intervals

Linear

📐 Linear Function

AROC is CONSTANT across all equal-length intervals.

The differences between consecutive AROCs are zero.

This constant AROC is just the slope of the line. Same from anywhere to anywhere.

Quadratic

📈 Quadratic Function

AROC changes at a CONSTANT RATE over consecutive equal-length intervals.

The differences between consecutive AROCs are constant (non-zero).

The AROCs form a linear pattern — they go up or down by the same amount each step.

Neither

🔀 Neither

AROC changes, but NOT at a constant rate.

The differences between consecutive AROCs are not constant.

Could be exponential, cubic, or some other type of function.

🎯

The 3-Step Method

Step 1: Verify all input intervals are equal (same Δx).
Step 2: Compute AROC = Δy/Δx for each consecutive interval.
Step 3: Check differences between consecutive AROCs.
   • Differences = 0 → Linear
   • Differences are constant (non-zero) → Quadratic
   • Differences not constant → Neither

✏️

Proving Not Linear — Example 1

Unequal input intervals: compute slope on each sub-interval

📌 Example 1 — The table gives selected values of f(x). Explain why f(x) is NOT a linear function.

x	f(x)
1	4
2	7
4	10
8	13

→

Interval	AROC = Δy/Δx
[1, 2]	(7−4)/(2−1) = 3
[2, 4]	(10−7)/(4−2) = 3/2
[4, 8]	(13−10)/(8−4) = 3/4

NOT linear — the AROCs are 3, 3/2, 3/4 — NOT constant. A linear function requires the same slope on every interval.

⚠️

Careful: unequal intervals here

In Example 1, the x-intervals are NOT equal (Δx = 1, 2, 4). This means we can't use the "AROC pattern" test for linear vs quadratic — we can only verify constancy of slope. Since the slopes differ, it's not linear. The linear/quadratic identification method requires equal Δx.

📊

Quadratic AROC Pattern — Example 2

g(x) = x² — equal intervals, AROCs form a linear sequence

📌 Example 2 — g(x) = x². Compute the AROC for each equal-length interval. What pattern do you notice?

x	g(x)
−3	9
−1	1
1	1
3	9
5	25

→

Interval (Δx=2)	AROC	Δ between AROCs
[−3, −1]	(1−9)/2 = −4	—
[−1, 1]	(1−1)/2 = 0	+4
[1, 3]	(9−1)/2 = 4	+4
[3, 5]	(25−9)/2 = 8	+4

AROCs: −4, 0, 4, 8 → increase by 4 each time → constant rate → QUADRATIC ✓

📋

Classify from Tables — Example 3

Linear · Quadratic · Neither — compute AROCs and check the pattern

a) f(x) Quadratic

x	f(x)	AROC
1	0	—
2	1	1/1 = 1
3	4	3/1 = 3
4	9	5/1 = 5

AROCs: 1, 3, 5 — increase by 2 each time (constant). Quadratic — AROC changes at a constant rate.

b) g(x) Linear

x	g(x)	AROC
1	0	—
2	1	1/1 = 1
5	4	3/3 = 1
10	9	5/5 = 1

Δx values: 1, 3, 5 (unequal). But AROC = Δy/Δx = 1/1 = 3/3 = 5/5 = 1 in each case. Constant AROC → Linear.

c) h(x) Quadratic

x	h(x)	AROC (Δx=2)
1	−1	—
3	1	2/2 = 1
5	2	1/2 = 1/2
7	2	0/2 = 0

AROCs: 1, 1/2, 0 — decrease by 1/2 each time (constant). Quadratic — AROC changes at a constant rate of −1/2.

〰️

Concavity and AROC — More on Concavity

The direction of AROC change determines concavity

∪

Concave Up

AROC is increasing over equal-length intervals. The slope is getting larger (more positive or less negative). Graph curves like a bowl.

∩

Concave Down

AROC is decreasing over equal-length intervals. The slope is getting smaller (less positive or more negative). Graph curves like a hill.

→

Neither (Linear)

AROC is constant — neither increasing nor decreasing. Graph is a straight line with no curve.

🔗

Connection between concavity and function type

Linear: AROC is constant → neither concave up nor concave down.
Quadratic (concave up): AROC is increasing → positive leading coefficient (parabola opens up).
Quadratic (concave down): AROC is decreasing → negative leading coefficient (parabola opens down).

📐

Concavity from Tables — Example 4

k, m, p — determine concave up, concave down, or neither

a) k(x) Concave Up

x	k(x)	AROC (Δx=0.1)
1	4	—
1.1	1	−3/0.1 = −30
1.2	−1	−2/0.1 = −20
1.3	−2	−1/0.1 = −10

AROCs: −30, −20, −10 — increasing (going from more negative to less negative). → Concave Up.

b) m(x) Neither

x	m(x)	AROC (Δx=0.1)
1	1	—
1.1	4	3/0.1 = 30
1.2	7	3/0.1 = 30
1.3	10	3/0.1 = 30

AROCs: 30, 30, 30 — constant. Linear function → no concavity → Neither.

c) p(x) Concave Down

x	p(x)	AROC (Δx=0.1)
1	1	—
1.1	7	6/0.1 = 60
1.2	11	4/0.1 = 40
1.3	13	2/0.1 = 20

AROCs: 60, 40, 20 — decreasing (going from larger to smaller). → Concave Down.

💡

Concave up even when decreasing!

In Example 4a, k(x) is decreasing (negative AROCs) yet concave up (AROCs increasing from −30 to −10). This is the key distinction from Topic 1.1 — a function can be decreasing AND concave up at the same time. Concavity is about whether the slope is getting bigger or smaller, not about whether the function itself is going up or down.

⚡

Quick Reference

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🔑 AROC Pattern → Function Type

Δ(AROC) = 0 → Linear

Constant slope = AROC never changes

Δ(AROC) = constant ≠ 0 → Quadratic

AROC changes at a constant rate

AROC increasing → Concave Up

AROC decreasing → Concave Down

⚠️ Common Mistakes

❌ Using unequal intervals to classify

The linear/quadratic AROC pattern test requires EQUAL input intervals (Δx must be the same for every step).

❌ Concave up means function is increasing

Concave up means AROC is increasing. The function itself can be decreasing and concave up simultaneously (Example 4a).

❌ Linear means neither concave up nor down

Correct! Linear → constant AROC → no concavity. "Neither" in the concavity context means linear.

❌ Checking only differences of outputs (not AROC)

For unequal Δx, you must compute AROC = Δy/Δx. Just looking at Δy will give wrong answers when intervals aren't equal.

Rates of Change:Linear & Quadratic

The 3-Step Method

Careful: unequal intervals here

Connection between concavity and function type

Concave up even when decreasing!

Rates of Change:
Linear & Quadratic