Unit 2: Exponential and Logarithmic Functions

Arithmetic and Geometric Sequences (Patterns Behind Linear vs. Exponential)

A sequence is an ordered list of numbers, where each number is a term. Sequences can be finite or infinite.

An arithmetic sequence is a sequence in which each successive term changes by a common difference, meaning a constant additive change (constant rate of change).

A geometric sequence is a sequence in which each successive term changes by a common ratio, meaning a constant multiplicative change (consistent proportional change).

This difference-vs.-ratio idea is one of the fastest ways to decide what kind of function model fits a situation. Over equal-length input intervals, if output values change at a constant rate (you keep adding the same amount), that points to a linear relationship. If output values change at a proportional rate (you keep multiplying by the same factor), that points to an exponential relationship. In many problems, you can determine the model type and parameters using just two distinct values from a sequence or two input-output points from a function.

Exam Focus
  • Typical question patterns:
    • Identify whether data behave more like an arithmetic pattern (constant differences) or geometric pattern (constant ratios).
    • Use two values to determine a linear vs. exponential relationship.
  • Common mistakes:
    • Checking differences when you should check ratios (or vice versa).
    • Forgetting that “proportional change” means multiplying by a constant factor, not adding a constant amount.

Exponential Functions: What They Are and How Their Graphs Behave

What an exponential function is (and why it’s different from a power function)

An exponential function is a function where the input variable appears in the exponent. The parent form is:

f(x)=b^x

Here, b is a positive constant called the base.

This matters because exponential functions naturally model repeated multiplication: the change in a quantity is proportional to the current amount (population growth, interest, radioactive decay, etc.). By contrast, a power function like:

g(x)=x^2

has the variable in the base, not the exponent. That swap completely changes the behavior and graph shape.

Growth vs. decay (the role of the base)

For the parent function b^x, the base controls whether the function grows or decays.

b>1

means exponential growth.

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