022 Exam 2 Sample B, Problem 2

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

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
$y\,=\,f(x),$
and we have a transformed graph
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=A\,f(x-B)+C,}
we would have to consider a shift/mirroring of the basic graph from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , a horizontal shift from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} , and a vertical shift from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} .

 Solution:

Step 1:
Identify the Basic Graph: The basic graph is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\left(\frac{1}{2}\right)^x.}
If you do not know exactly what this looks like, plot the basic points:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+1} is zero when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=-1} ). Note that since the basic graph has an asymptote at the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis, we will shift the asymptote to the line Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=-4.}

Final Answer:
In addition to your final graph, for grading purposes you should show your basic graph, the new asymptote and the translations of a few points. The red dots show the values for the basic graph from the chart in step 1. Many teachers will also ask that you label a few points on your final graph (shown here in blue).

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