Lesson 5.2

The Exponential Function

The function is the parent exponential function. Its graph always passes through , has a horizontal asymptote at , and never touches the x-axis.

Introduction

Every exponential function has two key parameters: (the initial value = y-intercept) and (the base = growth/decay factor). The base determines the shape; transformations shift it.

Past Knowledge

Growth vs. decay (5.1), exponent rules, transformations.

Today's Goal

Graph and apply transformations.

Future Success

The natural base (5.3) becomes the most important exponential function.

Key Concepts

Parent Graphs

Blue: · Purple: · Red:

Key Features

y-intercept: Always for since

Asymptote: (x-axis) — the curve gets close but never touches

Domain: — all real numbers

Range: — always positive

Worked Examples

Example 1: Plotting Points

Basic

Graph by plotting key points.

x	−2	−1	0	1	2	3
	¼	½	1	2	4	8

Rises steeply to the right, approaches 0 to the left

Example 2: Transformations

Intermediate

Graph and identify the asymptote.

Shifts: right 1, up 3

New y-intercept: . Asymptote shifts from to .

Asymptote:

Example 3: Reflection

Advanced

Graph and state the range.

The negative sign flips the graph over the x-axis

The parent rises; falls. Adding 4 shifts everything up 4.

Key points

, ,

Asymptote: · Range:

Common Pitfalls

Asymptote Shifts With Vertical Shift

has asymptote , not . The moves the asymptote up by .

Range Is Never All Reals

Exponential functions never output zero or negatives (unless reflected). The range is , never .

Real-Life Applications

Moore's Law (transistor count doubles every ~2 years) is exponential. Understanding explains why your phone is millions of times more powerful than 1970s computers.