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Use translations, reflections, and stretches to graph \(\displaystyle f(x) = - 8 e^{x - 3} - 1\). State the horizontal asymptote and the range of \(\displaystyle f(x)\).
The graph is shifted right 3 units and down 1 unit. Since the coefficient is negative the graph is reflected across the \(\displaystyle x-\)axis and stretched by a factor of 8. The horizontal asymptote is \(\displaystyle y= -1\). The range is \(\displaystyle (-\infty, -1)\)
\begin{tikzpicture}[scale=.5]
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\draw[latex-latex, very thick] (-10.4,0)--(10.4,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-10.4)--(0,10.4) node[above]{$y$};
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\draw[very thick, latex-latex, color=blue, samples = 200, domain=-10:3.1] plot ({\x}, {-8*exp(\x - 3) - 1});
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\begin{question}Use translations, reflections, and stretches to graph $f(x) = - 8 e^{x - 3} - 1$. State the horizontal asymptote and the range of $f(x)$.\newline
\soln{11cm}{The graph is shifted right 3 units and down 1 unit. Since the coefficient is negative the graph is reflected across the $x-$axis and stretched by a factor of 8. The horizontal asymptote is $y= -1$. The range is $(-\infty, -1)$\newline\begin{tikzpicture}[scale=.5]
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}
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\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\draw[very thick, latex-latex, color=blue, samples = 200, domain=-10:3.1] plot ({\x}, {-8*exp(\x - 3) - 1});
\draw[thick, dashed] (-10, -1) -- (10, -1);
\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) = - 8 e^{x - 3} - 1 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20-%208%20e%5E%7Bx%20-%203%7D%20-%201%20" alt="LaTeX: \displaystyle f(x) = - 8 e^{x - 3} - 1 " data-equation-content=" \displaystyle f(x) = - 8 e^{x - 3} - 1 " /> . 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 3 units and down 1 unit. 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 8. The horizontal asymptote is <img class="equation_image" title=" \displaystyle y= -1 " src="/equation_images/%20%5Cdisplaystyle%20y%3D%20-1%20" alt="LaTeX: \displaystyle y= -1 " data-equation-content=" \displaystyle y= -1 " /> . The range is <img class="equation_image" title=" \displaystyle (-\infty, -1) " src="/equation_images/%20%5Cdisplaystyle%20%28-%5Cinfty%2C%20-1%29%20" alt="LaTeX: \displaystyle (-\infty, -1) " data-equation-content=" \displaystyle (-\infty, -1) " /> <br><?xml version="1.0" encoding="UTF-8"?>
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