4.01 Transformations of exponential functions

Exponential functions and their graphs

A base form of an exponential function is $y=a^x$y=ax for $a>0$a>0 and $a\ne1$a≠1 and the variable $x$x is in the exponent. These graphs take the following form:

$a>1$a>1	$00<a<1
Key features: Exponential growth: As the $x$x-values increase, the $y$y-values increase at an increasing rate. The $y$y-intercept is $\left(0,1\right)$(0,1), since when $x=0$x=0, $y=a^0=1$y=a0=1, for any positive value $a$a. $y=0$y=0 is a horizontal asymptote for each graph. As $x\ \rightarrow-\infty$x →−∞, $y\ \rightarrow0^+$y →0+ Domain: $x$x is any real number Range: $y>0$y>0	Key features: Exponential decay: As the $x$x-values increase, the $y$y-values decrease at a decreasing rate. The $y$y-intercept is $\left(0,1\right)$(0,1), since when $x=0$x=0, $y=a^0=1$y=a0=1, for any positive value $a$a. $y=0$y=0 is a horizontal asymptote for each graph. As $x\ \rightarrow\infty$x →∞, $y\ \rightarrow0^+$y →0+ Domain: $x$x is any real number Range: $y>0$y>0

How did the graph of $y=\left(\frac{1}{2}\right)^x$y=(12​)x compare to that of $y=2^x$y=2x? Can you see they are a reflection of each other in the $y$y-axis? In general, for $a>0$a>0 the graph of $g\left(x\right)=\left(\frac{1}{a}\right)^x$g(x)=(1a​)x is equivalent to $g\left(x\right)=a^{-x}$g(x)=a−x, which is a decreasing exponential function and a reflection of the graph $f\left(x\right)=a^x$f(x)=ax in the $y$y-axis. 

Transformations of exponential functions

Let's look at the graphs of $f\left(x\right)=a^x$f(x)=ax and $y=A\times a^{b\left(x-h\right)}+k$y=A×ab(x−h)+k and the impact the parameters have on the key features. Use the applet below to observe the impact of $A$A, $b$b, $h$h and $k$k for a particular $a$a value:

We can see in particular, the vertical translation by $k$k units causes the horizontal asymptote to become $y=k$y=k.

Practice questions

Question 1

Question 2

Question 3