Sketch the graph of 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 + 1} - 4}
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| Foundations:
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| This is a problem about graphing through transformations. It requires you to find the basic or prototype graph, and then understand how to apple the transformations. In particular, if our basic graph is
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- 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\,=\,f(x),}
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| and we have a transformed graph
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- 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,}
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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  .
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Solution:
| Step 1:
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| Identify the Basic Graph: The basic graph is
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- 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.}
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| If you do not know exactly what this looks like, plot the basic points:
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- 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}}
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| 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.
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| Step 2:
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| Verify the Transformations: Here, we need to shift the basic graph down by four, while moving it to the right 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}
is one). 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.}
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| Final Answer:
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| Not done yet!
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