Mathfolis

Change in Linear and Exponential Functions

Unit 2 · Exponential and Logarithmic Functions

What AP Precalc asks here

Linear functions are additive: a unit step in x adds a fixed amount to y. Exponential functions are multiplicative: a unit step in x multiplies y by a fixed factor. The classic numerical fact is that exponential always wins the long-run race against linear — but linear can lead for small x. Difference tests detect linear; ratio tests detect exponential.

The two families

Linear
f(x)=mx+b(constant first difference)f(x) = mx + b \quad\text{(constant first difference)}
Exponential
f(x)=a⋅bx(constant ratio)f(x) = a \cdot b^x \quad\text{(constant ratio)}
AP Tip: If two models agree on three equally spaced points, compute a fourth point — linear and exponential typically diverge there.
Type 1

Difference test vs ratio test

Compute first differences. If constant, the data is linear. Otherwise compute successive ratios; if constant, it's exponential.

Example 1
Is the function f(0)=5,f(1)=10,f(2)=20,f(3)=40,f(4)=80f(0) = 5, f(1) = 10, f(2) = 20, f(3) = 40, f(4) = 80 linear or exponential? (A) Linear with slope 5 (B) Exponential with growth factor 2 (C) Linear with slope 15 (D) Neither

Practice more of this type— AI-generated · always-new problems

Generate Problems →
Type 2

Build both models from two points

From two points, find the linear model y=mx+by = mx + b and the exponential model y = a · bxb^x. Compare them at a third point.

Example 2
Both a linear and an exponential model pass through (0,4)(0, 4) and (2,16)(2, 16). Find the two models. (A) Linear f(x)=6x+4f(x) = 6x + 4; Exponential f(x)=4⋅2xf(x) = 4 \cdot 2^x (B) Linear f(x)=4x+4f(x) = 4x + 4; Exponential f(x)=4⋅4xf(x) = 4 \cdot 4^x (C) Linear f(x)=8x+4f(x) = 8x + 4; Exponential f(x)=4⋅2xf(x) = 4 \cdot 2^x (D) Linear f(x)=6x+4f(x) = 6x + 4; Exponential f(x)=4⋅4xf(x) = 4 \cdot 4^x

Practice more of this type— AI-generated · always-new problems

Generate Problems →
Type 3

Long-term comparison

Plug in the given x to each model and compare. For small x, linear can win; for large x, exponential always wins eventually.

Example 3
For f(x)=100xf(x) = 100x and g(x)=2xg(x) = 2^x, which is larger at x=5x = 5? (A) ff is larger (B) gg is larger (C) They are equal (D) Cannot be determined

Practice more of this type— AI-generated · always-new problems

Generate Problems →
Change in Linear and Exponential Functions | AP Precalculus — Mathfolis