# Exponential Functions and the Number e

> Growth and decay, graphs of exponentials, Euler's number e and the derivative of e to the x.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/exponential-functions/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The video that went viral
Olivia posts a funny video of her dog. On the first day, 3 friends share it. The next day each of them gets 3 more people to share it, and so on. Day 1: 3 shares. Day 2: 9. Day 3: 27. By day 10, it is 59 049.

Adding the same amount each day gives a straight line. Multiplying by the same amount each day gives **exponential growth**, which starts slowly and then explodes. Bank interest, bacteria, and the cooling of a hot drink all follow exponential rules.

**In short:** An exponential function has the variable in the power, like $y = 2^x$, so it grows or decays by the same factor each step. Its graph passes through $(0, 1)$ with the asymptote $y = 0$, and the special base $e \approx 2.718$ has its own gradient: $\dfrac{d}{dx}e^x = e^x$.

## Starting point: Years 9 and 10

### Index laws reminder

$$a^m a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn} \qquad a^0 = 1 \qquad a^{-n} = \frac{1}{a^n}$$

### Exponential functions

In an **exponential function**, the variable is in the power:

$$y = a^x \quad (a > 0, \; a \neq 1)$$

| $x$ | $-2$ | $-1$ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| $2^x$ | $\dfrac{1}{4}$ | $\dfrac{1}{2}$ | 1 | 2 | 4 | 8 |

### Key idea: Features of y = aˣ when a > 1
- Passes through $(0, 1)$ because $a^0 = 1$.
- Always positive. It never touches the x-axis.
- Horizontal asymptote $y = 0$.
- Increasing: it climbs faster and faster.

$y = \left(\dfrac{1}{2}\right)^x = 2^{-x}$ is the mirror image in the y-axis. It shows **decay**.

### Calculator tip: Powers and the number e
Use the power key for powers: $2^{-3}$ is `2 ^ -3`, and $16^{\frac{3}{4}}$ is `16 ^ ( 3 ÷ 4 )`. Brackets around a fractional power are essential.

The number $e$ is above the `ln` key: `SHIFT ln` gives $e^x$, so $e^1 \approx 2.718$ and $e^{0.5} \approx 1.649$.

### Growth and decay

Something that grows by a fixed **percentage** follows $A = P(1 + r)^n$. Something that shrinks by a fixed percentage follows $A = P(1 - r)^n$.

### Worked example: Car value
A 30 000 dollar car loses 15% of its value every year. After 4 years:

$$A = 30\,000(0.85)^4 \approx 15\,660 \text{ dollars}$$

### Calculator tip: Do not round the growth factor
For 1000 growing by 5% a year for 8 years, type `1000 × 1.05 ^ 8` in one go (about 1477.46). For decay by 12% a year, the factor is 0.88. Round only the final answer.

### Solving simple exponential equations

Write both sides with the same base, then match the powers.

$$3^{x+1} = 81 = 3^4 \;\Rightarrow\; x + 1 = 4 \;\Rightarrow\; x = 3$$

### Watch out: 2³ is not 2 × 3
$2^3 = 2 \times 2 \times 2 = 8$. Powers mean repeated multiplication.

**Mind map**

```text
Exponentials
  Index laws
    Add, subtract, multiply powers
  y = aˣ
    Through (0, 1)
    Asymptote y = 0
    a > 1 grows
    0 < a < 1 decays
  Percent change
    P(1 + r)ⁿ
    P(1 - r)ⁿ
  Number e
    About 2.718
    Gradient equals height
    d/dx eˣ = eˣ
```

### Year 11: Euler's number e
Draw tangents to $y = 2^x$ and $y = 3^x$ at $x = 0$. The gradient of $2^x$ there is about 0.69. The gradient of $3^x$ is about 1.10. Somewhere between 2 and 3 is a base whose gradient at $x = 0$ is exactly 1. That number is

$$e \approx 2.718\,28$$

The function $y = e^x$ has an amazing property: **at every point, its gradient equals its height.**

$$\frac{d}{dx}e^x = e^x$$

At $x = 2$, the curve is at height $e^2 \approx 7.39$, and its tangent has gradient $e^2$ too.

**Examples.** $\dfrac{d}{dx}(5e^x) = 5e^x$. $\dfrac{d}{dx}(e^x + x^2) = e^x + 2x$.

$e$ also appears as the limit of compound interest: $\left(1 + \dfrac{1}{n}\right)^n \to e$ as $n$ gets very large.

### Year 11: Transforming exponentials
$y = e^{-x}$ is decay. $y = 2e^x - 3$ is $e^x$ stretched by 2 and moved down 3, so its asymptote is $y = -3$ and its y-intercept is $2 - 3 = -1$.

### Year 12: Exponential models and more derivatives
Continuous growth or decay is modelled by

$$N = N_0 e^{kt}$$

$k > 0$ for growth and $k < 0$ for decay. $N_0$ is the amount at $t = 0$.

Its derivative is $\dfrac{dN}{dt} = kN_0e^{kt} = kN$: the rate of change is proportional to the amount. More bacteria make new bacteria faster.

The chain rule gives $\dfrac{d}{dx}e^{f(x)} = f'(x)e^{f(x)}$. For example, $\dfrac{d}{dx}e^{3x^2} = 6xe^{3x^2}$.

**Reading the percentage.** In $P = A(1.2)^t$ the population is multiplied by 1.2 each step, a 20% increase. In $P = A(1.25)^{-t} = A(0.8)^t$ it is multiplied by 0.8, a 20% decrease.

**Finding $k$ from a doubling time or half-life.** If an amount doubles every 5 years, $2 = e^{5k}$, so $k = \dfrac{\ln 2}{5} \approx 0.139$. If it halves every 8 hours, $\frac{1}{2} = e^{8k}$, so $k = \dfrac{\ln 0.5}{8} = -\dfrac{\ln 2}{8}$. Taking logs to solve is explained in the Logarithms chapter. An amount that halves every 8 hours falls to one eighth after three halvings, 24 hours.

**Models with a constant.** Cooling follows $T = Ae^{-kt} + C$, where $C$ is the room temperature it settles to. For $T = 300e^{-0.1t} + 20$: it starts at 320 and approaches 20. To find when $T = 80$, solve $300e^{-0.1t} = 60$. The rate of cooling is $\dfrac{dT}{dt} = -30e^{-0.1t}$.

## Quick summary

- In an exponential function the variable is the power: $y = a^x$.
- $y = a^x$ passes through $(0, 1)$ with asymptote $y = 0$. $a > 1$ grows, $0 < a < 1$ decays.
- Percentage growth: $A = P(1 + r)^n$. Percentage decay: $A = P(1 - r)^n$.
- Solve $a^{f(x)} = a^k$ by matching powers.
- $e \approx 2.718$ and $\dfrac{d}{dx}e^x = e^x$.

## Quiz

Try these questions from the chapter quiz.

1. What is $2^5$?
   - A. 10
   - B. 32
   - C. 25
   - D. 64
   - Answer: B. 32. $2 \times 2 \times 2 \times 2 \times 2 = 32$.

2. Simplify $\dfrac{5^7 \times 5^2}{5^4}$.
   - A. $5^{14}$
   - B. $5^5$
   - C. $5^{13}$
   - D. $5^3$
   - Answer: B. $5^5$. $7 + 2 - 4 = 5$.

3. Every graph $y = a^x$ with $a > 0$ passes through:
   - A. $(1, 0)$
   - B. $(0, 1)$
   - C. $(0, 0)$
   - D. $(1, 1)$
   - Answer: B. $(0, 1)$. $a^0 = 1$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/exponential-functions/#quiz