Precalculus · Unit 2: Change in linear and exponential functions · 14 min read · Updated 2026-05-11
Change in linear and exponential functions — AP Precalculus
AP Precalculus · Unit 2: Change in linear and exponential functions · 14 min read
1. Constant Change in Linear Functions★★☆☆☆⏱ 4 min
Linear functions have the general form $f(x) = mx + b$, where $m$ is the slope (constant rate of change) and $b$ is the $y$-intercept (output when $x=0$). The defining characteristic of a linear function is that over any fixed interval of input $\Delta x = h$, the *absolute change* in output $\Delta f$ is constant, regardless of the starting value of $x$.
Exam tip: When asked for change, always include the correct sign for decreasing functions. AP exam graders dock points for missing the sign in contextual problems.
2. Proportional Change in Exponential Functions★★★☆☆⏱ 4 min
Exponential functions have the general form $f(x) = ab^x$, where $a \neq 0$ is the initial value (output when $x=0$) and $b>0, b \neq 1$ is the base (growth/decay factor per 1-unit increase in $x$). The defining characteristic of an exponential function is that over any fixed interval of input $\Delta x = h$, the *relative (proportional) change* in output is constant, regardless of the starting input $x$.
Exam tip: Always distinguish between growth factor $b$ and relative change. If the question asks for percentage change, you need to calculate $b-1$, not leave your answer as $b$. This is one of the most common point-losing mistakes on this topic.
3. Comparing Growth Rates of Linear and Exponential Functions★★★☆☆⏱ 3 min
A common AP exam question asks you to compare the output or growth rate of a linear and an exponential function, or find when exponential growth first overtakes linear growth. The core rule is that for any increasing linear function $f(x) = mx + b$ with $m>0$ and any increasing exponential growth function $g(x) = ab^x$ with $a>0$ and $b>1$, exponential growth will eventually outpace linear growth. However, over short time horizons, linear growth can often be larger than exponential growth, so you must always evaluate both functions for the given input.
Exam tip: When asked for the first full period where exponential overtakes linear, always check the integer before your candidate value to confirm it is still smaller. AP MCQ distractors are often designed to match the candidate value without this check.
4. Classifying Functions from Tabular Data★★★☆☆⏱ 3 min
AP Precalculus exams often ask you to classify a function as linear or exponential from tabular data, using the defining change properties of each function type. To do this, check for constant absolute change over equal input intervals to confirm linearity, or constant relative change over equal input intervals to confirm exponentiality.
Common Pitfalls
Why: Students confuse the ratio of final to initial output with the proportional change from initial to final
Why: Students memorize the general rule that exponential outpaces linear and forget it only applies for sufficiently large input
Why: Students mix up the defining properties of linear and exponential functions
Why: Students forget that exponential growth compounds, so growth factors multiply across intervals
Why: Students focus on the magnitude of the change and forget the direction