Intro to Equations of Exponential Functions

An exponential function is a mathematical function of the form $$y = a(b)^x$$ , where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. It shows how a quantity grows or decays at a constant rate.

The base in an exponential function represents the growth factor. 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 growth factor is the base 'b' in the function $$y = a(b)^x$$ . For example, in $$y = 5(2)^x$$ , the growth factor is 2.

The general form of an exponential growth equation is $$y = a(b)^x$$ , where 'a' is the initial value and 'b' is the growth factor.

The initial value is the value of 'y' when 'x' is 0. In the equation $$y = a(b)^x$$ , the initial value is 'a'.

To write an exponential function that starts at a specific value 'a' with a growth factor 'b', use the form $$y = a(b)^x$$ .

Linear growth increases by a constant amount, while exponential growth increases by a constant percentage. In exponential growth, the rate of increase itself grows over time.

If the ratio of consecutive outputs (y-values) is constant, the function is exponential. For example, if the values double or triple consistently, it indicates exponential growth.

The standard form of an exponential function is $$y = a(b)^x$$ , where 'a' is the initial value and 'b' is the base or growth factor.

Identify the initial value and the growth factor from the problem. Then, use the form $$y = a(b)^x$$ to write the equation.