Exponential Graphs

An exponential function is a mathematical function of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base of the exponential (a positive real number), and 'x' is the exponent.

The graph of an exponential growth function rises rapidly as x increases, approaches the x-axis but never touches it (asymptote), and passes through the point (0, a).

The graph of an exponential decay function decreases rapidly as x increases, approaches the x-axis but never touches it (asymptote), and passes through the point (0, a).

An exponential growth function has a base 'b' greater than 1 in the equation f(x) = a * b^x.

An exponential decay function has a base 'b' between 0 and 1 in the equation f(x) = a * b^x.

The horizontal asymptote of an exponential function is the line y = 0, which the graph approaches but never reaches.

The base of an exponential function determines the rate of growth or decay; a base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.

The y-intercept of an exponential function f(x) = a * b^x is the point (0, a), where 'a' is the initial value.

Transformations such as vertical shifts, horizontal shifts, reflections, and stretches/compressions can change the position and shape of the graph of an exponential function.

Exponential functions grow or decay at a rate proportional to their current value, while linear functions grow at a constant rate.