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Use translations, reflections, and stretches to graph \(\displaystyle f(x) = 2 - 7 e^{x - 7}\). State the horizontal asymptote and the range of \(\displaystyle f(x)\).
The graph is shifted right 7 units and up 2 units. Since the coefficient is negative the graph is reflected across the \(\displaystyle x-\)axis and stretched by a factor of 7. The horizontal asymptote is \(\displaystyle y= 2\). The range is \(\displaystyle (-\infty, 2)\)
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\begin{question}Use translations, reflections, and stretches to graph $f(x) = 2 - 7 e^{x - 7}$. State the horizontal asymptote and the range of $f(x)$.\newline
\soln{11cm}{The graph is shifted right 7 units and up 2 units. Since the coefficient is negative the graph is reflected across the $x-$axis and stretched by a factor of 7. The horizontal asymptote is $y= 2$. The range is $(-\infty, 2)$\newline\begin{tikzpicture}[scale=.5]
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\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\draw[very thick, latex-latex, color=blue, samples = 200, domain=-10:7.5] plot ({\x}, {2 - 7*exp(\x - 7)});
\draw[thick, dashed] (-10, 2) -- (10, 2);
\end{tikzpicture}
}
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Use translations, reflections, and stretches to graph <img class="equation_image" title=" \displaystyle f(x) = 2 - 7 e^{x - 7} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%202%20-%207%20e%5E%7Bx%20-%207%7D%20" alt="LaTeX: \displaystyle f(x) = 2 - 7 e^{x - 7} " data-equation-content=" \displaystyle f(x) = 2 - 7 e^{x - 7} " /> . State the horizontal asymptote and the range of <img class="equation_image" title=" \displaystyle f(x) " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20" alt="LaTeX: \displaystyle f(x) " data-equation-content=" \displaystyle f(x) " /> .<br> </p> </p><p> <p>The graph is shifted right 7 units and up 2 units. Since the coefficient is negative the graph is reflected across the <img class="equation_image" title=" \displaystyle x- " src="/equation_images/%20%5Cdisplaystyle%20x-%20" alt="LaTeX: \displaystyle x- " data-equation-content=" \displaystyle x- " /> axis and stretched by a factor of 7. The horizontal asymptote is <img class="equation_image" title=" \displaystyle y= 2 " src="/equation_images/%20%5Cdisplaystyle%20y%3D%202%20" alt="LaTeX: \displaystyle y= 2 " data-equation-content=" \displaystyle y= 2 " /> . The range is <img class="equation_image" title=" \displaystyle (-\infty, 2) " src="/equation_images/%20%5Cdisplaystyle%20%28-%5Cinfty%2C%202%29%20" alt="LaTeX: \displaystyle (-\infty, 2) " data-equation-content=" \displaystyle (-\infty, 2) " /> <br><?xml version="1.0" encoding="UTF-8"?>
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