Exponential Functions

An exponential function is a mathematical function of the form $$f(x) = a \times b^x$$ , where $$a$$ is a constant, $$b$$ is a positive real number, and $$x$$ is the exponent.

The base of an exponential function determines the rate of growth or decay. If $$b > 1$$ , the function represents exponential growth; if $$0 < b < 1$$ , it represents exponential decay.

The y-intercept of an exponential function $$f(x) = a \times b^x$$ is the point where the graph intersects the y-axis, which occurs at $$f(0) = a$$ .

To find the y-intercept, evaluate $$f(0) = 2^0 = 1$$ . Thus, the y-intercept is (0, 1).

The initial value is the y-intercept, which is $$f(0) = 3^0 = 1$$ .

An asymptote is a line that the graph of a function approaches but never touches. For exponential decay functions, the horizontal asymptote is typically the x-axis (y = 0).

The range of this function is all positive real numbers, or $$y > 0$$ .

To determine the equation, identify the initial value and the base by analyzing the growth or decay pattern in the table.

The 'a' value represents the initial amount or starting value of the function when $$x = 0$$ .

The general form of an exponential decay function is $$f(x) = a \times b^x$$ , where $$0 < b < 1$$ .