Exponential Equations

An exponential equation is an equation in which a variable appears in the exponent. For example, $$y = a(b^x)$$ where a and b are constants.

The base of an exponential function determines the rate of growth or decay. If the base is greater than 1, the function represents exponential growth; if the base is between 0 and 1, it represents exponential decay.

The general form of an exponential function is $$f(x) = a(b^x)$$ where a is the initial value, b is the base, and x is the exponent.

Exponential growth is identified by a curve that rises steeply as x increases, indicating that the function's value increases rapidly.

Exponential decay is identified by a curve that falls steeply as x increases, indicating that the function's value decreases rapidly.

The starting point of an exponential function is the value of the function when x = 0, which is given by $$f(0) = a(b^0) = a$$ .

Linear functions have a constant rate of change, represented by a straight line, while exponential functions have a variable rate of change, represented by a curve.

An asymptote is a line that a graph approaches but never touches. For exponential decay, the x-axis (y=0) is a horizontal asymptote.

To write an exponential function from a table, identify the initial value and the growth/decay factor by examining the ratio of consecutive outputs.

The initial value (a) represents the starting amount before any growth or decay occurs, often found at x=0.