Mathfolis

Change in Tandem

Unit 1 · Polynomial and Rational Functions

What AP Precalc asks here

Two quantities change in tandem when one depends on the other — the foundation of function notation y = f(x). Topic 1.1 is qualitative: you read change from tables (first differences), from graphs (rises and falls), and from equations (rate-of-change reasoning). Concavity enters as a preview of AP Calculus: when the rate of change is itself increasing, f is concave up; when decreasing, f is concave down.

Increasing / decreasing on an interval I

Increasing
x1<x2  ⟹  f(x1)<f(x2) for all x1,x2∈Ix_1 < x_2 \implies f(x_1) < f(x_2) \text{ for all } x_1, x_2 \in I
Decreasing
x1<x2  ⟹  f(x1)>f(x2) for all x1,x2∈Ix_1 < x_2 \implies f(x_1) > f(x_2) \text{ for all } x_1, x_2 \in I

Concavity from rate of change

Concave up
rate of change of f is increasing\text{rate of change of } f \text{ is increasing}
Concave down
rate of change of f is decreasing\text{rate of change of } f \text{ is decreasing}
Caution: 'Increasing' is not the same as 'positive'. f(x) = 2x − 7 is increasing everywhere, but is only positive for x > 3.5.
Type 1

Read change from a table

Compare successive outputs in a table. If they rise across an interval, f is increasing there; if they fall, decreasing; if they stay the same, constant.

Example 1
A function f is given by the table: f(0) = 5, f(1) = 8, f(2) = 10, f(3) = 11, f(4) = 11. On what subinterval is f decreasing? (A) [0, 1] (B) [3, 4] (C) [0, 4] (D) No subinterval where f is decreasing

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Type 2

Read change from a graph or equation

For a linear function, the sign of the slope determines whether f increases or decreases everywhere. For a quadratic, the vertex separates the increasing and decreasing branches.

Example 2
Consider f(x)=2x−7f(x) = 2x - 7. On what interval is f increasing, and where is f positive? (A) Increasing on (3.5,∞)(3.5, \infty); positive on R\mathbb{R} (B) Increasing on R\mathbb{R}; positive on (3.5,∞)(3.5, \infty) (C) Increasing on R\mathbb{R}; positive on R\mathbb{R} (D) Increasing on (−∞,3.5)(-\infty, 3.5); positive on (3.5,∞)(3.5, \infty)

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Type 3

Concavity from rate of change

Compute first differences (or successive AROCs) of f. If they grow, f is concave up; if they shrink, concave down.

Example 3
The first differences of f over equally spaced inputs are 5,3,1,−15, 3, 1, -1. Is f concave up or concave down on this interval? (A) Concave up (B) Concave down (C) Linear (D) Cannot be determined from differences

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