4.2 Graphs of exponential functions  (Page 4/6)

Before graphing, identify the behavior and key points on the graph.

• Since $b=\frac{1}{2}$ is between zero and one, the left tail of the graph will increase without bound as $x$ decreases, and the right tail will approach the x -axis as $x$ increases.
• Since $a=4,$ the graph of $f(x)={\left(\frac{1}{2}\right)}^x$ will be stretched by a factor of $4.$
• Create a table of points as shown in [link] .

  $x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
  $$f(x)=4(\frac{1}{2}{)}^x$$	$32$	$16$	$8$	$4$	$2$	$1$	$0.5$

• Plot the y- intercept, $\left(0,4\right),$ along with two other points. We can use $\left(-1,8\right)$ and $\left(1,2\right).$

Draw a smooth curve connecting the points, as shown in [link] .

The domain is $\left(-\infty ,\infty \right);$ the range is $\left(0,\infty \right);$ the horizontal asymptote is $y=0.$

Since we want to reflect the parent function $f(x)={\left(\frac{1}{4}\right)}^x$ about the x- axis, we multiply $f(x)$ by $-1$ to get, $g(x)=-{\left(\frac{1}{4}\right)}^x.$ Next we create a table of points as in [link] .

Plot the y- intercept, $\left(0,\mathrm{-1}\right),$ along with two other points. We can use $\left(\mathrm{-1},\mathrm{-4}\right)$ and $\left(1,\mathrm{-0.25}\right).$

Draw a smooth curve connecting the points:

The domain is $\left(-\infty ,\infty \right);$ the range is $\left(-\infty ,0\right);$ the horizontal asymptote is $y=0.$