Exponential Functions
Lessons in this chapter
About Exponential Functions
Exponential Functions covers where exponential change comes from and how to build and read exponential models.
Sequences & Change Patterns shows how repeated multiplicative change in a sequence leads naturally to an exponential function.
Exponential Functions and Transformations covers graphing exponential growth and decay, plus the shifts and stretches that transform them.
Exponential Models closes the chapter by building and interpreting exponential models for real growth and decay situations.
The three lessons, Sequences & Change Patterns, Exponential Functions and Transformations, and Exponential Models, each open as a practice worksheet worked in the page and scored as you answer. The modeling problems are the ones worth repeating, since choosing between a linear and an exponential model is what the exam actually asks for. Free to practice online.
Frequently asked questions
Sequences & Change Patterns separates arithmetic growth, where a constant is added each step, from geometric growth, where a constant is multiplied. Exponential Functions and Transformations covers the shape of an exponential graph, its horizontal asymptote, and what shifting and stretching does to both. Exponential Models fits exponential functions to real situations such as growth, decay, and compounding. This sits in Functions (F) in the high school Common Core, and in Unit 2 of the AP Precalculus course framework, Exponential and Logarithmic Functions.
A sequence with a constant ratio between consecutive terms (repeated multiplicative change) grows exactly like an exponential function evaluated at whole-number inputs.
An exponential function has the input in the exponent, producing repeated multiplicative growth or decay rather than the repeated addition of a linear function.
Changing an exponential equation’s coefficients shifts, stretches, compresses, or reflects its graph, the same transformation ideas as any other function.
Identify the initial amount, the growth or decay factor, and the time unit from the real situation, then plug them into the exponential model’s general form.
Forgetting that a decay factor must be between 0 and 1: a factor greater than 1 describes growth, not decay, even if the exponent itself is written the same way.