Exponential Growth and Decay

Exponential growth occurs when the increase in a quantity is proportional to its current value, leading to growth at an increasing rate. It can be represented by the equation: $$y = a(1 + r)^t$$ , where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

Exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. It can be represented by the equation: $$y = a(1 - r)^t$$ , where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

The y-intercept of an exponential function is the value of 'y' when 'x' is 0. In the equation $$y = a(1 + r)^t$$ , the y-intercept is 'a'.

To write an equation for exponential decay, use the formula: $$y = a(1 - r)^t$$ , where 'a' is the initial amount and 'r' is the decay rate.

The formula for exponential growth is: $$y = a(1 + r)^t$$ , where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

The growth factor for a population that doubles every year is 2.

A decay factor of 0.7 means that the quantity decreases to 70% of its previous value each time period.

The number of periods can be determined by the time variable 't' in the equations $$y = a(1 + r)^t$$ for growth and $$y = a(1 - r)^t$$ for decay.

The base in 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.

Exponential growth is represented by a curve that rises steeply, while exponential decay is represented by a curve that falls steeply.