Exponential functions
Doubling, halving, interest earning interest. What kind of rule puts the variable up in the exponent, and what does its graph look like?
Where you are 6 inside
What this area is about
In a linear rule the input is multiplied by a fixed slope. In an exponential function f(x) = a \cdot b^x (with a \ne 0, b > 0 and b \ne 1) the input is the exponent, so each step of 1 in x multiplies the output by the same factor b. This area builds those rules, draws their graphs, and meets the special base e.
How these fit together
The idea. Exponential growth and decay is its own area: growth and decay factors, doubling time and half life, and races against linear and polynomial growth.
Graphs. Graphs of exponential functions draws y = b^x, which passes through (0, 1) and runs flat along the x-axis on one side. Transforming an exponential graph shifts and stretches it, and moves its horizontal asymptote with it.
A procedure. Writing an exponential function from a table or graph reads the starting value a and the constant ratio b back off the data.
Money and a new base. Compound interest splits a yearly rate over several periods. Letting the number of periods grow without end leads to the natural exponential function y = e^x, built on the number e.
A path through it
Growth and decay first, then the graph. After the graph, write a rule from a table or graph and transform the graph, in either order. Compound interest also follows growth and decay, and e comes after compound interest.
Next, logarithms undo exponentials (once you also have inverse functions), exponential equations can be solved, and growth rates at infinity shows b^x with b > 1 outgrowing every power of x.