2.2 Change in Linear and Exponential Functions

Linear functions ($f(x) = b + mx$) work like arithmetic sequences: you start with an initial value and repeatedly add a constant rate of change. Exponential functions ($f(x) = ab^x$) work like geometric sequences: you start with an initial value and repeatedly multiply by a constant factor. In AP Precalculus, this comparison helps you decide whether data changes by a constant difference or a constant ratio.

Why This Matters for the AP Precalculus Exam

This topic connects the sequences you studied in 2.1 to the continuous functions that fill the rest of Unit 2 and beyond. Being able to switch between linear and exponential thinking sets you up for exponential modeling, regressions, and comparing competing models later in the unit. On the AP Precalculus exam, you may see this idea on both multiple-choice and free-response questions, where you decide whether data changes at a constant rate (linear) or proportionally (exponential), then build the matching function. Showing clear steps when you set up and solve for a slope or base is important for clear exam work, since answers without supporting work may not support a stronger score on free-response.

more resources to help you study

practice multiple choiceFRQ practice & scoringcheatsheetsscore calculator

Key Takeaways

• A linear function $f(x) = b + mx$ mirrors an arithmetic sequence $a_n = a_0 + dn$: same initial value plus repeated addition of a constant.
• An exponential function $f(x) = ab^x$ mirrors a geometric sequence $g_n = g_0 r^n$: same initial value times repeated multiplication by a constant.
• Over equal-length input intervals, constant change in output means linear; proportional change in output means exponential.
• Point-based forms let you build either function from a known point: $f(x) = y_i + m(x - x_i)$ for linear, $f(x) = y_i r^{(x - x_i)}$ for exponential.
• Any of these four (arithmetic sequence, linear function, geometric sequence, exponential function) can be pinned down from two distinct values.
• A sequence and its matching function can have different domains: sequences live on whole numbers, while functions are usually defined on all real numbers.

Arithmetic Sequence Lookalikes: Linear Functions

A linear function has the form $f(x) = b + mx$, where $b$ is the y-intercept (the point where the line crosses the y-axis) and $m$ is the slope (the constant rate of change). A positive slope means the line rises, a negative slope means it falls, and a slope of 0 means it is horizontal.

Recall that an arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. It can be written as $a_n = a_0 + dn$, where $a_0$ is the initial term and $d$ is the common difference. You generate the sequence by adding $d$ to the previous term.

Both linear functions and arithmetic sequences can be described as an initial value plus repeated addition of a constant rate of change. In a linear function, the initial value is the y-intercept $b$ and the constant rate of change is the slope $m$. In an arithmetic sequence, the initial value is the first term $a_0$ and the constant rate of change is the common difference $d$.

Because of this shared structure, both can model situations with a constant rate of change.

Point-Based Form for Linear Functions

Just like arithmetic sequences, linear functions can also be built from a known point and a constant rate of change. In an arithmetic sequence, the known value is the kth term $a_k$ and the rate of change is the common difference $d$. In a linear function, the known value is a point $(x_i, y_i)$ and the rate of change is the slope $m$.

The point-slope form of a linear function, $f(x) = y_i + m(x - x_i)$, and the explicit formula of an arithmetic sequence, $a_n = a_k + d(n - k)$, both let you find a value from one known value plus a constant rate of change.

The formulas look alike, but they describe different things: linear functions are continuous functions, while arithmetic sequences are discrete sequences.

Geometric Sequence Lookalikes: Exponential Functions

An exponential function has the variable $x$ in the exponent rather than the base. Its general form is $f(x) = ab^x$, where $a$ is the initial value and $b$ is the base (a positive number not equal to 1). Depending on the value of $b$, this models exponential growth or decay.

Recall that a geometric sequence is a list of numbers where the ratio between consecutive terms is always the same. It can be written as $g_n = g_0 r^n$, where $g_0$ is the initial term and $r$ is the common ratio. You generate the sequence by multiplying the previous term by $r$.

Both exponential functions and geometric sequences can be described as an initial value times repeated multiplication by a constant proportion. In an exponential function, the initial value is $a$ and the constant proportion is the base $b$. In a geometric sequence, the initial value is the first term $g_0$ and the constant proportion is the common ratio $r$.

Because of this, both can model situations with exponential growth or decay, such as compound interest, population growth, and radioactive decay.

Point-Based Form for Exponential Functions

Like geometric sequences, exponential functions can also be built from a known value and a constant ratio. In a geometric sequence, the known value is the kth term $g_k$ and the ratio is the common ratio $r$. In an exponential function, the known value is a point $(x_i, y_i)$ and the ratio is $r$.

The point-based form of an exponential function, $f(x) = y_i r^{(x - x_i)}$, and the explicit formula of a geometric sequence, $g_n = g_k r^{(n - k)}$, both let you find a value from one known value plus a constant ratio.

Again, similar-looking formulas describe different things: functions are not the same as sequences.

Caveats of Sequence vs. Function: Domains

Sequences and their matching functions may have different domains. The domain of a sequence is typically the whole numbers, while the domain of a function can be any set of real numbers. For example, the sequence $a_n = 2n$ has a domain of whole numbers, but the corresponding function $f(x) = 2x$ has a domain of all real numbers.

In the image below, notice how the function (red line) includes values on the negative x-axis, while the sequence (dots) is limited to what comes at or after 0.

The same idea applies to exponential cases. The sequence $a_n = (1/2)^n$ has a domain of whole numbers, while the corresponding function $f(x) = (1/2)^x$ is defined for all real numbers.

A sequence can also be defined with a recursive formula, which gives the nth term in terms of earlier terms. In that case the sequence's domain may be limited to the whole numbers for which the formula works. A corresponding function can cover a wider domain using an explicit formula that connects the term to the input variable.

Similarities and Differences

1. Over equal-length input-value intervals, look at how the output values change:

• Change at a constant rate: the function is linear. The difference between output values stays the same across equal input steps.
• Change proportionally: the function is exponential. The ratio between output values stays the same across equal input steps.

2. Both forms use an initial value and a constant involved with change.

• In a linear function $f(x) = b + mx$, the initial value is the y-intercept $b$ and the constant is the slope $m$.
• In an exponential function $f(x) = ab^x$, the initial value is $a$ and the constant is the base $b$.

The big difference: linear functions are based on addition, while exponential functions are based on multiplication.

3. Arithmetic sequences, linear functions, geometric sequences, and exponential functions can each be determined by two distinct values.

• Arithmetic sequence: a known term plus the common difference (or two terms).
• Linear function: a point plus the slope (or two points).
• Geometric sequence: a known term plus the common ratio (or two terms).
• Exponential function: a point plus the base (or two points).

How to Use This on the AP Precalculus Exam

MCQ

When you see a table of values, check the change between equal input steps before picking an answer. If the outputs go up by the same amount each step, the data is linear and you want $f(x) = b + mx$. If the outputs are multiplied by the same factor each step, the data is exponential and you want $f(x) = ab^x$. Confusing constant difference with constant ratio is the most common trap here.

Problem Solving

To build a function from two points $(x_1, y_1)$ and $(x_2, y_2)$:

• For a linear function, find the slope $m = \frac{y_2 - y_1}{x_2 - x_1}$, then use point-slope form $f(x) = y_i + m(x - x_i)$.
• For an exponential function, find the ratio with $b = \left(\frac{y_2}{y_1}\right)^{\frac{1}{x_2 - x_1}}$, then plug a known point into $f(x) = ab^x$ to solve for $a$.

Write your steps clearly when solving for a slope or base. Answers without supporting work may not support a stronger score on free-response.

Common Trap

Watch the domain. If a problem describes a sequence (whole-number inputs only), do not treat it as a continuous function, and do not connect the points into a smooth curve unless the problem actually gives you a function of the real numbers.

Common Misconceptions

• Constant difference is not the same as constant ratio. Linear functions add the same amount each step; exponential functions multiply by the same factor each step. Check which one a data set actually shows before choosing a model.
• A sequence and its matching function are not identical. They may share a formula structure, but sequences are defined on whole numbers (discrete points), while functions are usually defined on all real numbers (a smooth curve).
• The base $b$ in $f(x) = ab^x$ is a multiplier, not a slope. It tells you the factor of change per unit, not a constant amount added.
• You do not need a full table to build these functions. Two distinct values are enough to determine a linear function, an exponential function, an arithmetic sequence, or a geometric sequence.
• In point-slope and point-based forms, $(x_i, y_i)$ is a known point, not the initial value at $x = 0$. Do not assume $y_i$ is the y-intercept unless $x_i = 0$.

Related AP Precalculus Guides

• 2.9 Logarithmic Expressions
• 2.11 Logarithmic Functions
• 2.5 Exponential Function Context and Data Modeling
• 2.8 Inverse Functions
• 2.12 Logarithmic Function Manipulation
• 2.4 Exponential Function Manipulation