Exponential growth and decay

Exponential graphs y = a^x, growth and decay models A = A₀(1 ± r)^n, half-life and doubling, and using trial and error or graphs to find when a value is reached.

Exponential graphs

y = b^x

Every graph passes through (0, 1) and approaches the x-axis (an asymptote) without touching it.

Growth and decay models

Exponential model

$$A = A_0(1 + r)^n \text{ (growth)} \qquad A = A_0(1 - r)^n \text{ (decay)}$$

Example 1●●

A town of $2000$ grows by $5\%$ per year. Find the population after $3$ years.

$2000 \times 1.05^3 = 2315.25$

About $2315$ people

Example 2●●

A substance has a half-life of $5$ years. How much of $80$ g remains after $15$ years?

$15$ years is $3$ half-lives.

$80 \times (\tfrac{1}{2})^3 = 10$ g

Flashcards

Growth factor for 5% growth

$1.05$

Half-life

Time for half to decay

Asymptote of $y = 2^x$

$y = 0$ (the $x$-axis)

Practice

$2000$ grows $5\%$ per year. Population after $3$ years (nearest whole)?
Half-life $5$ years. Grams of $80$ g left after $15$ years?
$y$-intercept of $y = 3^x$?
$y = (0.5)^x$ is:
Bacteria double every hour, starting with $100$. After how many hours are there $1600$?
What is the multiplier for $8\%$ decay per year?
A $\$40\,000$ car loses $20\%$ of its value each year. Value after $4$ years?
The asymptote of $y = 2^x$ is: