Writing & Transformations - Exponentials

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

Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to rapid increases over time, typically represented by a function of the form $$y = a \times b^x$$ with $$b > 1$$ .

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

The base of an exponential function $$y = a \times b^x$$ is the number $$b$$ that is raised to the power of $$x$$ .

The parent function of exponential functions is $$y = b^x$$ , where $$b > 0$$ and @@b eq 1@@.

Adding a constant to the exponent translates the graph horizontally. For example, $$y = b^{x + c}$$ translates the graph left by $$c$$ units if $$c > 0$$ .

Subtracting a constant from the exponent translates the graph horizontally. For example, $$y = b^{x - c}$$ translates the graph right by $$c$$ units if $$c > 0$$ .

Multiplying an exponential function by a constant affects the vertical stretch or compression of the graph. For example, $$y = a \times b^x$$ stretches the graph vertically if $$a > 1$$ and compresses it if $$0 < a < 1$$ .

To write an exponential function from a table, identify the initial value (when $$x = 0$$ ) and the growth/decay factor by examining the ratio of consecutive values.

Translating an exponential function up or down involves adding or subtracting a constant from the entire function. For example, $$y = b^x + k$$ translates the graph up by $$k$$ units if $$k > 0$$ .