Exponential Functions

An exponential function is a mathematical function of the form $$y = a \times b^x$$ , where $$a$$ is a constant, $$b$$ is a positive real number, and $$x$$ is the exponent. It describes growth or decay processes.

The base of an exponential function indicates the growth or decay rate. If the base is greater than 1, the function represents exponential growth. If the base is between 0 and 1, it represents exponential decay.

Adding a constant to an exponential function translates the graph vertically. For example, in $$y = b^x + k$$ , the graph shifts up by $$k$$ if $$k > 0$$ and down if $$k < 0$$ .

Subtracting a constant from an exponential function translates the graph vertically down. For example, in $$y = b^x - k$$ , the graph shifts down by $$k$$ .

Exponential growth occurs when a quantity increases at a rate proportional to its current value, typically represented by a function of the form $$y = a \times b^x$$ where $$b > 1$$ .

Exponential decay occurs when a quantity decreases at a rate proportional to its current value, typically represented by a function of the form $$y = a \times b^x$$ where $$0 < b < 1$$ .

You can identify exponential growth from a function by checking if the base of the exponent is greater than 1. For example, in $$y = 2^x$$ , the base is 2, indicating growth.

You can identify exponential decay from a function by checking if the base of the exponent is between 0 and 1. For example, in $$y = (1/2)^x$$ , the base is 1/2, indicating decay.

The horizontal asymptote of an exponential function is a line that the graph approaches but never touches. For functions of the form $$y = a \times b^x + k$$ , the horizontal asymptote is $$y = k$$ .

The graph of an exponential growth function rises steeply to the right and approaches the horizontal asymptote to the left.