Growth

Understanding Exponential Growth

In the chapter on Exponential Functions, you saw the general idea of functions of the form
$$
y = a \cdot b^x
$$
where $a$ is an initial value and $b$ is a constant base. In this chapter, we focus specifically on growth: situations where the quantity increases as time (or some other variable) increases.

For exponential growth, the key condition is
$$
b > 1.
$$

Here, $x$ usually represents time or number of steps, and $y$ is the size of the quantity we are tracking.

Typical features:

• The quantity grows faster and faster (not at a constant rate).
• The graph curves upward and becomes steeper as $x$ increases.
• Equal time steps correspond to equal percentage increases, not equal amount increases.

Basic Formulas for Exponential Growth

Discrete-time growth: repeated multiplication

Many growth processes are described step-by-step, such as “each year the population increases by 5%.” If something starts at value $P_0$ (read “P-zero”) and grows by the same percentage rate each time period, the general formula is
$$
P(t) = P_0 (1 + r)^t
$$
where:

• $P_0$ is the initial amount,
• $r$ is the growth rate per period (written as a decimal, e.g. $5\% = 0.05$),
• $t$ is the number of periods (years, months, etc.),
• $P(t)$ is the amount after $t$ periods.

Why $1 + r$?

• “No change” would be multiplying by $1$.
• “Increase by $r$” means adding $r$ times the current amount.
• So “new = old + r \cdot \text{old} = (1 + r)\cdot\text{old}$.

Examples of interpreting $1 + r$

• $10\%$ growth per year $\Rightarrow r = 0.10$, base $b = 1 + 0.10 = 1.10$.
• $3\%$ growth per month $\Rightarrow r = 0.03$, base $b = 1.03$.
• $0.5\%$ growth per day $\Rightarrow r = 0.005$, base $b = 1.005$.

In each case, $b > 1$, which means exponential growth.

Continuous-time growth with $e$

In many applications, especially in more advanced math and science, growth is written using the constant $e$:
$$
P(t) = P_0 e^{kt}
$$
where $k$ is the continuous growth rate. For growth, we have $k > 0$.

In this form:

• If $k > 0$, we have exponential growth.
• If $k < 0$, we have exponential decay (covered in the Decay chapter).

You do not need to master the connection between $k$ and $r$ here, but the important idea is:

• $P(t) = P_0(1+r)^t$ and $P(t) = P_0 e^{kt}$ both describe exponential growth.
• They just use different ways of writing the base (a general $b>1$ or a special base $e$).

This chapter will mostly use the $P(t) = P_0(1+r)^t$ form, which is more common in basic applications and word problems.

Recognizing Exponential Growth from a Formula

You are dealing with exponential growth when all of the following are true:

• The variable (often time) appears in the exponent.
• The base of the exponent is greater than 1.
• The function has the general shape
  $$
  y = a \cdot b^x \quad\text{with } a>0,\, b>1.
  $$

Examples of growth functions:

• $A(t) = 500 \cdot 1.08^t$ (8% growth per period)
• $P(t) = 2 \cdot 3^t$ (tripling each period)
• $N(t) = 1200 \cdot e^{0.04t}$ (continuous growth at rate $k = 0.04$)

Non-examples (not exponential growth):

• $y = 5x + 7$ (linear, not exponential)
• $y = 200 \cdot 0.7^t$ (exponential decay, because $0.7 < 1$)
• $y = 3^x - 10$ with negative outputs only because $a<0$ (this might grow in magnitude but behaves differently and is usually not used to model growth of a positive quantity).

Reading Growth Rate from a Formula

Suppose you have an exponential growth function already written. How do you find the growth rate?

For a function in the form
$$
P(t) = P_0 (1 + r)^t,
$$
the growth rate is just $r$.

• If $P(t) = 400(1.12)^t$, then $r = 0.12$, i.e. $12\%$ growth per period.
• If $P(t) = 2500(1.005)^t$, then $r = 0.005$, i.e. $0.5\%$ growth per period.

If your function is written as $P(t) = ab^t$, then:

• The growth factor per period is $b$.
• The rate $r$ is $b-1$.
• The percentage growth rate is $(b-1)\times 100\%$.

Example:

• $P(t) = 1000 \cdot 1.03^t$

Here $b = 1.03$,

so $r = 1.03 - 1 = 0.03 = 3\%$.

For the $e$-form:
$$
P(t) = P_0 e^{kt},
$$
the constant $k$ is the continuous growth rate. Translating $k$ to a percentage growth per unit time depends on context and is typically handled in more advanced discussions. For now, you can treat $k$ similarly to $r$ in the sense that:

• $k > 0$ means growth.
• Larger $k$ means faster growth.

Graph Features of Exponential Growth

For $y = a b^x$ with $a > 0$ and $b > 1$:

• Domain: all real $x$.
• Range: $y > 0$.
• $y$-intercept: at $x = 0$, we have $y = a b^0 = a$, so the graph passes through $(0, a)$.
• Horizontal asymptote: the $x$-axis (the line $y = 0$) is a horizontal asymptote; the graph gets closer to $y = 0$ as $x$ becomes very negative.
• Behavior:
• As $x \to +\infty$, $y \to +\infty$ (it shoots upward).
• As $x \to -\infty$, $y \to 0^+$ (approaches zero from above).
• Shape: A smooth curve increasing, with the slope getting steeper as $x$ increases.

Changing parameters:

• Increasing $a$ shifts the graph upward (and changes the $y$-intercept).
• Changing $b$ (while $b>1$) changes how fast the graph rises:
• Larger $b$ $\Rightarrow$ steeper growth.
• Smaller $b$ (closer to 1) $\Rightarrow$ slower growth.

Comparing Exponential Growth to Linear Growth

In many problems, you compare exponential growth to linear change.

• Linear: adds a fixed amount each period.
• Exponential: multiplies by a fixed factor each period.

For example, suppose:

• Plan A: savings account increases by \$50 each year:
  $$
  A(t) = 100 + 50t
  $$
• Plan B: savings grows by $5\%$ each year:
  $$
  B(t) = 100(1.05)^t
  $$

At first, the linear plan may give more money (because \$50 is bigger than the early 5% increases). But as time goes on, the exponential plan will eventually overtake the linear plan, because the percentage increase applies to a growing amount.

This “eventually overtakes” behavior is characteristic of exponential growth: it starts out slow but grows very fast for large $t$.

Typical Word Problems About Growth

Here are the main types of questions you encounter with exponential growth formulas. The exact solving steps will be covered in detail in other chapters (e.g., solving equations), but the structure here is specific to growth.

1. Given initial value and rate, find future value

General form:
$$
P(t) = P_0 (1 + r)^t.
$$

Typical wording:

• “A population of 2000 increases by 4% each year. How many will there be after 10 years?”
• “An investment of \$500 grows by 6% per year. What is it worth after 15 years?”

In both cases, you:

• Identify $P_0$ and $r$,
• Plug in $t$,
• Compute $P(t)$.

2. Finding the growth rate from data

Typical wording:

• “A population grows from 1000 to 1210 in 2 years, and is assumed to grow exponentially. What is the annual growth rate?”

Idea:

• Use $P(t) = P_0(1 + r)^t$,
• Plug in $P_0$, $P(t)$, and $t$,
• Solve for $r$.

You might write
$$
1210 = 1000(1 + r)^2
$$
and then solve the equation for $r$.

3. Time to reach a given size

Typical wording:

• “A bacteria culture starts with 500 cells and doubles every 3 hours. How long until there are 8000 cells?”
• “\$2000 is invested at $5\%$ interest compounded yearly. How long until it doubles?”

You:

• Write the exponential growth model.
• Plug in the desired final amount for $P(t)$.
• Solve for $t$.

This often involves logarithms (covered in the Logarithms chapter), but the problem setup—the part specific to growth—is choosing the correct exponential form and placing $P_0$, $r$, and $t$ correctly.

Common Real-World Models of Exponential Growth

Exponential growth appears in many contexts. Here we focus on the algebraic form and the interpretation of parameters, not the full scientific details.

Population growth (simple model)

A basic model:
$$
P(t) = P_0 (1 + r)^t
$$

• $P_0$: initial population.
• $r$: growth rate per period (often per year).
• $t$: time in that period.

Interpretation:

• Each period, the population increases by the same percentage, not the same number of individuals.

Compound interest

For money growing with interest, a very common model is:
$$
A(t) = P\left(1 + \frac{r}{n}\right)^{nt}
$$
where:

• $P$ is the initial principal (starting amount).
• $r$ is the annual interest rate (decimal).
• $n$ is the number of compounding periods per year (e.g. $12$ for monthly).
• $t$ is time in years.
• $A(t)$ is the amount after $t$ years.

Notice that:

• The base is $1 + \dfrac{r}{n} > 1$, so this is exponential growth in $t$.
• When $n$ is fixed, this has the general exponential form in $t$.

If interest is compounded once per year ($n = 1$), this reduces to:
$$
A(t) = P(1 + r)^t
$$
which is exactly the same form as the basic growth formula.

Doubling time

Many exponential growth situations can be described by a doubling time: the time it takes for the quantity to double.

If a quantity doubles every $d$ units of time, you can model it as:
$$
P(t) = P_0 \cdot 2^{t/d}
$$

• After $t = d$, you get $P(d) = P_0 \cdot 2^{1} = 2P_0$.
• After $t = 2d$, you get $P(2d) = P_0 \cdot 2^{2} = 4P_0$.
• After $t = 3d$, you get $P(3d) = P_0 \cdot 2^{3} = 8P_0$, etc.

This is another standard way of writing exponential growth, emphasizing how long it takes to double instead of giving the growth rate per period.

You can rewrite this in the form $P_0(1+r)^t$ by recognizing that
$$
2^{t/d} = \left(2^{1/d}\right)^t
$$
so the base $b = 2^{1/d}$, and the growth factor per one time unit is $2^{1/d}$.

Interpreting Growth in Tables and Graphs

Sometimes you are not given an explicit formula. Instead, you might see a table of values or a graph. Here is how to recognize and interpret exponential growth in such representations.

From a table

If $x$ increases by 1 (or a fixed amount) each row, and the $y$-values are being multiplied by the same factor each time, then $y$ is growing exponentially with respect to $x$.

Example table:

• $x: 0,\; 1,\; 2,\; 3,\; 4$
• $y: 5,\; 7.5,\; 11.25,\; 16.875,\; 25.3125$

Check ratios:

• $\dfrac{7.5}{5} = 1.5$,
• $\dfrac{11.25}{7.5} = 1.5$,
• $\dfrac{16.875}{11.25} = 1.5$,
• $\dfrac{25.3125}{16.875} = 1.5$.

Each step multiplies by $1.5$. So we have exponential growth with:

• Initial value $a = 5$,
• Growth factor $b = 1.5$,
• Growth rate $r = 0.5 = 50\%$ per step.

So the function is
$$
y = 5(1.5)^x.
$$

From a graph

On a graph, exponential growth:

• Starts slowly (near the $x$-axis for negative $x$).
• Crosses the $y$-axis at $(0, a)$.
• Rises more and more steeply for larger $x$.
• Never touches or crosses the horizontal asymptote $y=0$.

If you zoom in near large $x$, the graph can appear “almost vertical,” but it is always a smooth curve, not a straight line.

Summary of Key Ideas About Growth

• Exponential growth occurs when a quantity repeatedly multiplies by a constant factor greater than 1.
• Standard discrete-time growth model:
  $$
  P(t) = P_0(1 + r)^t, \quad r > 0.
  $$
• The base $1+r$ is the growth factor per period, and $r$ is the growth rate.
• In $P(t) = ab^t$ with $b > 1$:
• Growth factor: $b$,
• Growth rate: $r = b - 1$.
• The graph has:
• Domain: all real $x$,
• Range: positive $y$,
• Horizontal asymptote at $y = 0$,
• $y$-intercept at $(0,a)$,
• Increasing and getting steeper as $x$ increases.
• Exponential growth is used to model populations, compound interest, repeated percentage increases, and any process where equal time steps bring equal percentage changes.