022 Exam 2 Sample B, Problem 2

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Sketch the graph of ${\displaystyle y=\left({\frac {1}{2}}\right)^{x+1}-4}$.

Foundations:
This is a problem about graphing through transformations. It requires you to find the basic or prototype graph, and then understand how to apply the transformations. In particular, if our basic graph is
${\displaystyle y\,=\,f(x),}$
and we have a transformed graph
${\displaystyle y=A\,f(x-B)+C,}$
we would have to consider a vertical scaling/mirroring of the basic graph from ${\displaystyle A}$, a horizontal shift from ${\displaystyle B}$, and a vertical shift from ${\displaystyle C}$.

Solution:

Step 1:
Identify the Basic Graph: The basic graph is
${\displaystyle y=\left({\frac {1}{2}}\right)^{x}.}$
If you do not know exactly what this looks like, plot the basic points:
${\displaystyle {\begin{array}{|c||c|c|c|c|c|}\hline x&-2\,&-1\,&\,\,0\,\,&1&2\\\hline (1/2)^{x}&\,4&\,2&1&1/2&1/4\\\hline \end{array}}}$
I would always recommend plotting the basic graph, in order to show that you properly applied the transformations. Note that since our base is less than one, the basic graph will be decreasing.
Step 2:
Verify the Transformations: Here, we need to shift the basic graph down by four, while moving it to the left one (as the argument ${\displaystyle x+1}$ is zero when ${\displaystyle x=-1}$). Note that since the basic graph has an asymptote at the ${\displaystyle x}$-axis, we will shift the asymptote to the line ${\displaystyle y=-4.}$