Exponential Growth and Decay

Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to a rapid increase over time. It can be represented by the function $$y = a(1 + r)^t$$ , where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

Exponential decay is the process of reducing an amount by a consistent percentage rate over a period of time. It can be represented by the function $$y = a(1 - r)^t$$ , 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 the function when $$x = 0$$ . It is found by evaluating the function at this point, typically represented as $$f(0) = a$$ .

To identify the percent growth from an exponential equation of the form $$y = a(1 + r)^x$$ , the growth rate 'r' is expressed as a percentage by multiplying it by 100.

The base of an exponential function indicates the growth or decay 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.

To write an exponential function for population growth, use the formula $$P(t) = P_0(1 + r)^t$$ , where $$P_0$$ is the initial population, $$r$$ is the growth rate, and $$t$$ is time.

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

To determine if a function represents exponential growth or decay, examine the base of the exponential function. If the base is greater than 1, it indicates growth; if the base is between 0 and 1, it indicates decay.

The initial value in an exponential function represents the starting amount before any growth or decay occurs. It is denoted as 'a' in the function $$y = a(1 + r)^t$$ .

To solve an exponential equation graphically, plot the function on a graph and find the point where it intersects with the line representing the other side of the equation. The x-coordinate of the intersection point gives the solution.