Exponential Function

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 @@b eq 1@@. It describes growth or decay processes.

The general form of an exponential growth function is $$f(x) = a \times (1 + r)^x$$ , where $$r > 0$$ is the growth rate.

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

A function is exponential growth if the base of the exponent is greater than 1 (e.g., $$f(x) = 3 \times (2)^x$$ ).

A function is exponential decay if the base of the exponent is between 0 and 1 (e.g., $$f(x) = 3 \times (1/2)^x$$ ).

Reflecting an exponential function across the x-axis changes the sign of the function, resulting in a downward-opening graph.

Shifting a function left by 3 units means replacing $$x$$ with $$x + 3$$ in the function's equation.

Shifting a function down by 6 units means subtracting 6 from the entire function's output.

The exponential function is $$f(x) = 400(1 + 0.03)^x$$ .

A negative coefficient reflects the graph across the x-axis, indicating a decay rather than growth.