Exponential Functions

Video lesson · from 1:27:112 worked examples

One lesson video covers all of Functions & Graphs, so it opens at 1:27:11 for this topic — not from the beginning.

Theory

Introduction

An exponential function is a function of the form $f(x) = a^x$ where $a \in \mathbb{R}$ and $a > 0$ .

Therefore, $f(x)$ is an exponential function to the base $a$ .

When $x = 0, f(0) = a^0 = 1$ . Graph passes through $(0, 1)$ .
When $x = 1, f(1) = a^1 = a$ . Graph passes through $(1, a)$ .

Every exponential graph of the form $f(x) = a^x$ passes through the points $(0, 1)$ and $(1, a)$ .

The Exponential Constant

$e$ can be considered one of the most important numbers in mathematics and is often called Euler's number. It is known to over 1 trillion digits of accuracy.

e = 2.718281828459045...

The Natural Exponential Function

The function $f(x) = e^x$ is called the natural exponential function.

⚠️ Common Examiner Traps

Every exponential passes through $(0,1)$ : because $a^0 = 1$ for any base. It is the single most useful checkpoint on a sketch.
Growth or decay depends on the base: $a \gt 1$ grows, $0 \lt a \lt 1$ decays. Sketching the wrong direction throws away the whole question.
The $x$ -axis is an asymptote: the curve approaches it but never reaches it, so there is no $x$ -intercept. Draw it approaching, not touching.
Transformations move the asymptote: adding a constant shifts the horizontal asymptote up or down with the curve.

Worked examples

Example 1

Transformations of Exponential Graphs

Given any graph $f(x)$ , we can reflect, move or scale it. The same rules apply for transforming exponential graphs as we have previously studied.

a) Sketch the graph of $y = 1 + 2^x$ .

b) Sketch the graph of $y = 3^{2x}$ .

a) Start with base graph $y = 2^x$ passing through $(0,1)$ and $(1,2)$ .

The $+1$ outside shifts the graph UP by 1 unit.

New key points: $(0,2)$ and $(1,3)$ . Asymptote moves to $y=1$ .

b) Start with base graph $y = 3^x$ passing through $(0,1)$ and $(1,3)$ .

The $2x$ inside squashes the graph horizontally by factor of 1/2.

Key points map: $(1,3) \to (0.5, 3)$ .

Example 2

Missing Constants

Part of the graph of $y = a^x + b$ is shown.

It passes through $(0, 2)$ and $(3, 9)$ .

Find $a$ and $b$ .

Substitute the point $(0, 2)$ into the equation:

2 = a^0 + b

2 = 1 + b \implies b = 1

Now equation is $y = a^x + 1$ . Substitute point $(3, 9)$ :

9 = a^3 + 1

a^3 = 8 \implies a = 2

So $a = 2$ , $b = 1$ .