Change in Linear and Exponential Functions

students, in this lesson you will learn how to describe and compare change in two important families of functions: linear and exponential 📈. This topic matters because many real situations grow or shrink over time, like saving money, spreading a virus, cooling coffee, or walking at a steady pace. By the end of this lesson, you should be able to explain how linear and exponential changes work, use the right vocabulary, and decide which model fits a situation better.

Lesson goals:

Explain what change means for linear and exponential functions.
Identify key terms such as rate of change, growth factor, and percent change.
Compare constant additive change with constant multiplicative change.
Use examples and calculations to connect these ideas to AP Precalculus.
See how this lesson fits inside the larger unit on exponential and logarithmic functions.

Understanding Change in a Function

A function describes how one quantity depends on another. In many AP Precalculus problems, the main question is not just “What is the output?” but “How does the output change when the input changes?” That is the heart of this lesson, students.

For a function $f(x)$, change can be measured by comparing two outputs. If the input changes from $x_1$ to $x_2$, the average rate of change is

$$\frac{f(x_2)-f(x_1)}{x_2-x_1}$$

This formula tells you the change in output per unit change in input. For linear functions, this value is constant. For exponential functions, the additive change is not constant, but the multiplicative change is.

A helpful way to think about it is this:

Linear functions add the same amount each step.
Exponential functions multiply by the same factor each step.

That difference is the key idea in this lesson ✨.

Linear Functions: Constant Additive Change

A linear function can be written in slope-intercept form as

$$f(x)=mx+b$$

where $m$ is the slope and $b$ is the initial value. The slope tells the constant rate of change. Every time $x$ increases by $1$, the output changes by $m$.

For example, suppose a student saves $20$ dollars each week. If $S(w)$ is the amount saved after $w$ weeks, then

$$S(w)=20w+50$$

if the student started with $50$. Each week, the total increases by $20$. The change is always additive:

Week 1 to Week 2: add $20$
Week 2 to Week 3: add $20$
Week 3 to Week 4: add $20$

The average rate of change is the same on every interval because the slope is constant.

Why linear change is easy to predict

Linear change is steady and uniform. If a taxi charges a $4$ dollar start fee plus $2$ dollars per mile, the cost increases by the same amount for each additional mile. If you graph it, the points lie on a straight line. This makes linear models useful for simple situations where change happens at a constant rate, such as hourly wages, fixed-distance travel, or constant filling rates.

Example: linear average rate of change

Let $f(x)=3x+2$. Find the average rate of change from $x=1$ to $x=5$.

$$\frac{f(5)-f(1)}{5-1}=\frac{17-5}{4}=\frac{12}{4}=3$$

The average rate of change is $3$, which matches the slope. Because the function is linear, this result would be the same for any interval.

Exponential Functions: Constant Multiplicative Change

An exponential function has the form

$$f(x)=ab^x$$

where $a$ is the initial value and $b$ is the growth factor. If $b>1$, the function shows growth. If $0<b<1$, the function shows decay.

The key idea is that each time $x$ increases by $1$, the output is multiplied by the same factor $b$. That means the percentage change stays constant, even though the actual amount of change gets larger or smaller.

For example, if a population of bacteria doubles every hour, then the growth factor is $2$. If there are $100$ bacteria at time $0$, the model is

$$P(t)=100\cdot 2^t$$

After $1$ hour, the population is $200$. After $2$ hours, it is $400$. After $3$ hours, it is $800$. Notice that the increases are $100$, then $200$, then $400$. The additive change is not constant, but the multiplication by $2$ is constant.

Growth factor and percent change

The growth factor is closely connected to percent change. If a quantity increases by $p\%$ each step, the growth factor is

$$1+\frac{p}{100}$$

For example, a $5\%$ increase corresponds to the factor

$$1+\frac{5}{100}=1.05$$

So a savings account with $5\%$ annual growth can be modeled as

$$A(t)=A_0(1.05)^t$$

If the account begins with $1000$, then

$$A(t)=1000(1.05)^t$$

Each year, the account is multiplied by $1.05$. The dollar increase gets bigger over time because the base amount is growing.

Example: exponential average rate of change

Let $g(x)=2^x$. Find the average rate of change from $x=0$ to $x=3$.

$$\frac{g(3)-g(0)}{3-0}=\frac{8-1}{3}=\frac{7}{3}$$

This is not a constant value for all intervals. For example, from $x=1$ to $x=2$,

$$\frac{g(2)-g(1)}{2-1}=\frac{4-2}{1}=2$$

Because exponential functions do not have constant additive change, their average rate of change depends on the interval.

Comparing Linear and Exponential Change

students, the biggest difference is this:

Linear change adds the same amount each step.
Exponential change multiplies by the same factor each step.

This difference creates very different graphs and real-world behavior.

Table comparison

Suppose both functions start at $100$.

Linear model:

$$L(t)=100+20t$$

Exponential model:

$$E(t)=100(1.2)^t$$

Let’s compare values.

At $t=0$: $L(0)=100$, $E(0)=100$
At $t=1$: $L(1)=120$, $E(1)=120$
At $t=2$: $L(2)=140$, $E(2)=144$
At $t=3$: $L(3)=160$, $E(3)=172.8$

At first, the outputs look similar. But over time, the exponential function pulls ahead because it keeps multiplying by $1.2$. This is why exponential growth can become much larger than linear growth over long periods.

Real-world example: money and interest

If you save money with a fixed deposit each month, that is often linear. If you invest money with compound interest, that is exponential. Compound interest is based on repeated multiplication, so the amount grows faster over time than simple addition.

For example, simple interest on $P$ dollars at rate $r$ for $t$ years is

$$A=P(1+rt)$$

This is linear in $t$.

Compound interest is

$$A=P\left(1+\frac{r}{n}\right)^{nt}$$

This is exponential. Because interest is earned on the current balance, the account grows faster as time passes.

Real-world example: medicine or population

If a quantity grows by a fixed percent each period, exponential models are often appropriate. Populations, bacterial colonies, and some investments can follow this pattern. If a quantity increases by a fixed amount each period, a linear model may be better, such as filling a tank at a constant rate or driving at a steady speed.

How AP Precalculus Uses These Ideas

On AP Precalculus problems, students, you may be asked to interpret a table, graph, equation, or description and decide whether the change is linear or exponential. To do this, look for evidence.

Clues for a linear model

A situation is likely linear if:

The differences between consecutive outputs are constant.
The graph is a straight line.
The equation can be written as $f(x)=mx+b$.
The rate of change is the same on every interval.

Clues for an exponential model

A situation is likely exponential if:

The ratios between consecutive outputs are constant.
The graph curves upward for growth or decreases toward zero for decay.
The equation can be written as $f(x)=ab^x$.
The quantity changes by the same percent each step.

Example: deciding from a table

Suppose a table shows values $4$, $8$, $16$, $32$ as $x$ increases by $1$ each time. The differences are $4$, $8$, and $16$, which are not constant. But the ratios are all $2$. That means the function is exponential with growth factor $2$.

Now suppose values are $7$, $11$, $15$, $19$. The differences are all $4$, so the function is linear with slope $4$.

Connecting to logarithms

Logarithmic functions are the inverse of exponential functions. That means logs help answer questions like: “What exponent gives this output?” For example, if

$$2^x=16$$

then

$$x=4$$

and this can also be written as

$$\log_2(16)=4$$

This connection matters because exponential and logarithmic functions are studied together in AP Precalculus. Understanding change in exponentials helps you understand why logarithms are useful for solving for time, exponents, and growth rates.

Conclusion

The main idea of this lesson is simple but powerful: linear and exponential functions describe change in two different ways. Linear functions change by adding the same amount each step, while exponential functions change by multiplying by the same factor each step. students, once you can tell the difference between constant differences and constant ratios, you can identify the model, interpret real-world situations, and prepare for more advanced work with exponential and logarithmic functions. This skill is important across the AP Precalculus unit because it builds the foundation for solving growth, decay, and inverse-function problems.

Study Notes

A function’s change can be measured using average rate of change:

$$\frac{f(x_2)-f(x_1)}{x_2-x_1}$$

Linear functions have the form $f(x)=mx+b$.
In a linear function, the rate of change is constant.
Linear change is additive: the same amount is added each step.
Exponential functions have the form $f(x)=ab^x$.
In an exponential function, the growth factor is constant.
Exponential change is multiplicative: the same factor is used each step.
Constant differences usually indicate a linear model.
Constant ratios usually indicate an exponential model.
A percent increase of $p\%$ corresponds to the factor $1+\frac{p}{100}$.
Exponential functions are closely connected to logarithmic functions because logs are their inverses.
In AP Precalculus, you should be able to identify, compare, and interpret linear and exponential change from equations, tables, graphs, and contexts.