Exponential Growth and Decay

By the end of this lesson, you will be able to read $a$ and $b$ out of an exponential function, tell growth from decay instantly, convert a percentage into a factor, and describe the graph including the line it never touches.

The Variable Moves into the Exponent

$$f(x) = a \cdot b^x \qquad a \neq 0, ; b > 0, ; b \neq 1$$

Every function so far had the variable in the base: $x^2$, $x^3$. Here it is in the exponent, and that swap changes the behavior completely.

$a$ is the initial value, because $f(0) = a \cdot b^0 = a$
$b$ is the factor: what the output gets multiplied by each time $x$ goes up by one

$$f(x) = 100 \cdot 2^x$$

Start at $100$, and double every step: $100, 200, 400, 800$.

The restrictions all earn their place. $b > 0$ because a negative base gives nonsense at fractional exponents; $b \neq 1$ because $1^x$ is the constant function $a$; $a \neq 0$ because $0 \cdot b^x$ is zero forever.

Constant MULTIPLICATIVE change is the definition, in the same way that a linear function has constant additive change. Add the same amount each step and you get a line; multiply by the same amount each step and you get this.

Growth or Decay Is Just b

$$b > 1 \quad \Rightarrow \quad \textbf{growth}$$

$$0 < b < 1 \quad \Rightarrow \quad \textbf{decay}$$

Multiplying by something bigger than $1$ makes the value climb; multiplying by a fraction shrinks it. The dividing line is exactly $1$, which is why $b = 1$ is excluded: it neither grows nor decays.

$$f(x) = 5 \cdot 3^x \quad \text{triples each step, growth}$$

$$f(x) = 80 \cdot (0.5)^x \quad \text{halves each step, decay}$$

Decay never reaches zero. Half of half of half is always something. The values approach zero and never arrive, which is why the graph has a horizontal asymptote rather than an $x$-intercept.

Sort It: Growth or Decay

Look only at $b$.

The last two matter most in practice. $b = 1.02$ and $b = 0.98$ look almost identical and behave oppositely, and the difference is entirely which side of $1$ they sit on.

Example 1: Reading a Function

Here is every step, worked all the way through. $$f(x) = 200 \cdot (1.5)^x$$

By the end of this lesson, you will be able to read $a$ and $b$ out of an exponential function, tell growth from decay instantly, convert a percentage into a factor, and describe the graph including the line it never touches.

The Variable Moves into the Exponent

$$f(x) = a \cdot b^x \qquad a \neq 0, ; b > 0, ; b \neq 1$$

Every function so far had the variable in the base: $x^2$, $x^3$. Here it is in the exponent, and that swap changes the behavior completely.

$a$ is the initial value, because $f(0) = a \cdot b^0 = a$
$b$ is the factor: what the output gets multiplied by each time $x$ goes up by one

$$f(x) = 100 \cdot 2^x$$

Start at $100$, and double every step: $100, 200, 400, 800$.

Growth or Decay Is Just b

$$b > 1 \quad \Rightarrow \quad \textbf{growth}$$

$$0 < b < 1 \quad \Rightarrow \quad \textbf{decay}$$

Multiplying by something bigger than $1$ makes the value climb; multiplying by a fraction shrinks it. The dividing line is exactly $1$, which is why $b = 1$ is excluded: it neither grows nor decays.

$$f(x) = 5 \cdot 3^x \quad \text{triples each step, growth}$$

$$f(x) = 80 \cdot (0.5)^x \quad \text{halves each step, decay}$$

Decay never reaches zero. Half of half of half is always something. The values approach zero and never arrive, which is why the graph has a horizontal asymptote rather than an $x$-intercept.

Sort It: Growth or Decay

Look only at $b$.

The last two matter most in practice. $b = 1.02$ and $b = 0.98$ look almost identical and behave oppositely, and the difference is entirely which side of $1$ they sit on.

Example 1: Reading a Function

Here is every step, worked all the way through. $$f(x) = 200 \cdot (1.5)^x$$

In words	$r$	$b$
grows $7%$ per year	$0.07$	$1.07$
grows $50%$ per year	$0.5$	$1.5$
falls $3%$ per year	$0.03$	$0.97$
falls $20%$ per year	$0.2$	$0.8$
doubles	$1$	$2$
halves	$0.5$	$0.5$

The Variable Moves into the Exponent

Growth or Decay Is Just b

Sort It: Growth or Decay

Example 1: Reading a Function

Exponential Growth and Decay

The Variable Moves into the Exponent

Growth or Decay Is Just b

Sort It: Growth or Decay

Example 1: Reading a Function

Percentages Become Factors

Worked Example 1: From Words to a Function

The Shape of the Graph

Plot It: The First Three Points

Worked Example 2: Doubling

Worked Example 3: Half-Life

Common Mistakes

Where This Goes Next

Check Your Understanding

Summary