Exponential Growth and Decay

Exponential growth occurs when a quantity increases by a consistent percentage over a period of time, resulting in a rapid increase. The general form is $$y = a(1 + r)^t$$ , where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

Exponential decay occurs when a quantity decreases by a consistent percentage over a period of time, leading to a rapid decrease. The general form is $$y = a(1 - r)^t$$ , where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

In a function of the form $$g(x) = a \times b^x$$ , if 'b' > 1, it represents exponential growth; if '0 < b < 1', it represents exponential decay.

The y-intercept of an exponential function $$g(x) = a \times b^x$$ is the value of 'g(0)', which equals 'a'.

Percent growth can be calculated using the formula: Percent Growth = (b - 1) * 100%, where 'b' is the base of the exponential function in the form $$g(x) = a \times b^x$$ .

Percent decay can be calculated using the formula: Percent Decay = (1 - b) * 100%, where 'b' is the base of the exponential function in the form $$g(x) = a \times b^x$$ .

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

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

The constant 'a' represents the initial value or the y-intercept of the exponential function.

To graph an exponential function, plot the y-intercept (a), identify the growth or decay based on the base, and plot additional points by substituting values for 'x'.