Exponential Growth

When something multiplies by the same factor repeatedly, the growth accelerates.

In 2020, scientists tracked the early spread of COVID-19 in a city where the case count doubled roughly every three days. On day one there were about 10 confirmed cases. By day nine there were around 80. By day fifteen, over 300. The count was not climbing by the same amount each day — it was multiplying by the same factor each period. That pattern is exponential growth.

The model for exponential growth is

f(x) = a \cdot b^x

where $a$ is the starting value (the output when $x = 0$ ), $b$ is the growth factor (a number greater than 1), and $x$ is the number of time periods that have passed. The reason $b$ must be greater than 1 is that each step multiplies the previous output by $b$ — if that factor is bigger than 1, the function grows.

To see why $a$ is the starting value, plug in $x = 0$ : you get $f(0) = a \cdot b^0 = a \cdot 1 = a$ . The starting value is always $a$ , regardless of what $b$ is.

Here is a concrete example. A social media post gets 5 shares on the first hour, and the number of shares triples every hour after that. The model is $f(x) = 5 \cdot 3^x$ , where $x$ is hours after the first hour.

Evaluating f(x) = 5 · 3^x at several values

f(0) = 5 \cdot 3^0 = 5 \cdot 1 = 5

At x = 0, the exponent is 0. Any nonzero number raised to 0 is 1, so you're left with just a.

f(1) = 5 \cdot 3^1 = 5 \cdot 3 = 15

After one hour: 5 tripled once.

f(2) = 5 \cdot 3^2 = 5 \cdot 9 = 45

After two hours: 5 tripled twice. Notice the jump from 15 to 45 is bigger than the jump from 5 to 15 — this is what acceleration looks like.

f(4) = 5 \cdot 3^4 = 5 \cdot 81 = 405

After four hours, you're in the hundreds. The function grows faster as x increases.

The graph of an exponential growth function has a recognizable shape: it starts close to the horizontal axis on the left, passes through the point $(0, a)$ , and then curves sharply upward to the right. It never crosses or touches the x-axis — as $x$ goes further left (more negative), the output gets smaller and smaller but stays positive. That boundary the graph approaches but never reaches is called a horizontal asymptote, and for $f(x) = a \cdot b^x$ it is always the x-axis, the line $y = 0$ .

Interactive graph — scroll to zoom, drag to pan

The parameter $a$ stretches the graph vertically — a larger $a$ lifts the entire curve, but the shape stays the same. The parameter $b$ controls how steep the curve gets. A larger $b$ means faster multiplication each step, so the curve climbs more aggressively. Changing $b$ while keeping $a$ fixed shifts how quickly the curve bends upward, but the y-intercept stays at $(0, a)$ .

Now a problem where you build the model from a description, then use it.

A colony of bacteria starts with 200 cells. The population doubles every hour. Write the exponential model and find the population after 5 hours.

Building and using a bacteria growth model

a = 200,\quad b = 2

The starting value is 200. Doubling means multiplying by 2 each period, so the growth factor is 2.

f(x) = 200 \cdot 2^x

Plug a and b into the general model.

f(5) = 200 \cdot 2^5

x = 5 because we want the population after 5 hours.

f(5) = 200 \cdot 32

2 to the fifth power is 32. It helps to know small powers of 2: 2, 4, 8, 16, 32.

f(5) = 6400 \checkmark

After 5 hours, the colony has 6,400 cells.

Sometimes the Regents gives you a table of values and asks you to identify whether it represents exponential growth and, if so, find $a$ and $b$ . The test for exponential growth in a table: check whether each output is obtained by multiplying the previous output by a constant ratio. If the ratio is the same throughout and greater than 1, you have exponential growth. Read $a$ directly from the output when $x = 0$ , and read $b$ from the constant ratio.

For the table below — $x = 0, 1, 2, 3$ with outputs $4, 12, 36, 108$ — check the ratios: $12 \div 4 = 3$ , $36 \div 12 = 3$ , $108 \div 36 = 3$ . Constant ratio of 3, starting value of 4. The model is $f(x) = 4 \cdot 3^x$ .

Practice Questions

f(x) = 2 \cdot 4^x,\quad \text{find } f(3)

f(x) = 50 \cdot \left(\frac{3}{2}\right)^x,\quad \text{find } f(2)

x: 0,\ 1,\ 2,\ 3 \quad f(x): 7,\ 21,\ 63,\ 189 \quad \text{Write the model.}

Regents Corner

On Part I and Part II of the Algebra I Regents, exponential growth questions often give you a model in function notation and ask you to interpret what $a$ or $b$ represents in context. For full credit on a Part II or III question, a numeric answer is not enough — you must state what it means. If the model is $f(x) = 300 \cdot 1.08^x$ and the question asks what 300 represents, write "300 is the initial value" or "300 is the starting population when $x = 0$ ." Writing just "300" earns no credit.

Students read b = 1.08 and write that the quantity grows by 1.08 each year, treating it as addition instead of multiplication. The quantity grows by a factor of 1.08, meaning it is multiplied by 1.08 — which is an 8% increase, not an increase of 1.08 units. These are very different. A population of 500 growing by a factor of 1.08 reaches 540 after one year, not 501.08.

← Previous

Exponential Decay

Geometric Sequences