Lesson 5.1

Growth vs. Decay

Linear functions grow by adding the same amount each step. Exponential functions grow by multiplying by the same factor. This simple difference creates dramatically different behavior.

Introduction

Imagine you have $100. Adding $10 each year is linear. Getting 10% interest each year is exponential. After 50 years, the linear approach gives $600 — the exponential gives over $11,000. That's the power of multiplication.

Past Knowledge

Linear functions, slope, exponent rules from Unit 1.

Today's Goal

Distinguish linear from exponential patterns and identify growth vs. decay.

Future Success

5.2 formalizes the graph of , and 5.4–5.5 apply these to finance.

Key Concepts

Linear vs. Exponential

FeatureLinearExponential
PatternAdd Multiply by
Formula
Graph shapeStraight lineCurve
ConstantRate of changeRatio

Growth vs. Decay

Growth:

Each step multiplies by more than 1 → values increase. Example: population doubling ().

Decay:

Each step multiplies by less than 1 → values decrease. Example: half-life ().

Worked Examples

Example 1: Identify the Pattern

Basic

Is the table linear or exponential?

x0123
y51545135
1

Check differences vs. ratios

Differences: 10, 30, 90 — not constant. Ratios: — constant!

Exponential with →

Example 2: Growth vs. Decay

Intermediate

Classify:

1

Identify

Since , this is decay. The value loses 15% each step.

Exponential Decay — 15% decrease per step

Example 3: Comparison Graph

Visual

Compare (linear) vs. (exponential).

Exponential (orange) eventually dominates linear (blue)

Common Pitfalls

Decay ≠ Negative

Decay means , not . An exponential function with a negative base oscillates — it's not simple decay.

Checking Differences on Exponential

Exponential data has a constant ratio, not constant differences. Always check both to classify correctly.

Real-Life Applications

Bacteria doubling every 20 minutes, radioactive decay, and viral social media posts all follow exponential patterns. Recognizing "constant ratio" data is one of the most practically useful math skills.

Practice Quiz

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