AP Precalculus · Unit 2: Exponential and Logarithmic Functions ·  Lesson 2.2

Change in Linear and Exponential Functions

Filling in the gaps between the dots  ·  Approx. 2 class days

Starringf(x) = b + mxf(x) = abx
Unit 2: Exponential and Logarithmic Functions

Use this as a quick reference for how linear functions match arithmetic sequences, how exponential functions match geometric sequences, and how to tell the two apart.

Change in Linear and Exponential Functions infographic

🧭 Plot Summary

In Lesson 2.1 you worked with sequences, which only exist at the whole numbers. This lesson fills in the gaps. A linear function f(x) = b + mx is an arithmetic sequence that is defined for every real number: an initial value plus repeated addition of a constant rate of change. An exponential function f(x) = abx is a geometric sequence that is defined for every real number: an initial value times repeated multiplication by a constant ratio. The big question is how to tell them apart. Look at equal-length input intervals. If the outputs change by a constant amount, the function is linear. If the outputs change proportionally, it is exponential.

What you'll do in this lesson

  • Connect arithmetic sequences to linear functions: both are an initial value plus repeated addition of a constant rate of change.
  • Write a linear function from a slope and any point, using f(x) = yi + m(x − xi).
  • Connect geometric sequences to exponential functions: both are an initial value times repeated multiplication by a constant ratio.
  • Write an exponential function from a ratio and any point, using f(x) = yibx − xi.
  • Explain why a sequence and its function have different domains.
  • Use equal-length input intervals to decide whether outputs change by a constant amount (linear) or by a constant factor (exponential).
  • Compare the families, then build either one from two distinct input-output values.

Why it matters

Real quantities do not only change on whole-number days or years. A phone loses value continuously, a savings account can be checked at any moment, and a cooling drink has a temperature at every instant. Linear and exponential functions are the two basic models for that kind of change, and Topic 2.3 will take a close look at the exponential one. Knowing which one fits a situation is the skill this lesson builds.

✅ Self-Check Before You Move On

Check off each item as you get there. These aren't grades — they're your own signal.