8-2 Transformations of Exponential Functions

Property

A vertical translation of an exponential function shifts the graph up or down by adding or subtracting a constant $k$ outside the base:

Property

A horizontal translation of an exponential function shifts the graph left or right. Starting from the parent function $f(x) = b^x$, a horizontal translation produces:

Property

For an exponential function of the form $f(x) = ab^x$, the value of $a$ determines a vertical dilation (stretch or compression) and, when $a < 0$, a reflection across the $x$-axis.

• If $|a| > 1$, the graph is vertically stretched by a factor of $|a|$.
• If $0 < |a| < 1$, the graph is vertically compressed by a factor of $|a|$.
• If $a < 0$, the graph is reflected across the $x$-axis.