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Use translations, reflections, and stretches to graph \(\displaystyle f(x) = - 5 e^{x + 6} - 3\). State the horizontal asymptote and the range of \(\displaystyle f(x)\).
The graph is shifted left 6 units and down 3 units. Since the coefficient is negative the graph is reflected across the \(\displaystyle x-\)axis and stretched by a factor of 5. The horizontal asymptote is \(\displaystyle y= -3\). The range is \(\displaystyle (-\infty, -3)\)
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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:-5.7] plot ({\x}, {-5*exp(\x + 6) - 3});
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\begin{question}Use translations, reflections, and stretches to graph $f(x) = - 5 e^{x + 6} - 3$. State the horizontal asymptote and the range of $f(x)$.\newline
\soln{11cm}{The graph is shifted left 6 units and down 3 units. Since the coefficient is negative the graph is reflected across the $x-$axis and stretched by a factor of 5. The horizontal asymptote is $y= -3$. The range is $(-\infty, -3)$\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:-5.7] plot ({\x}, {-5*exp(\x + 6) - 3});
\draw[thick, dashed] (-10, -3) -- (10, -3);
\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) = - 5 e^{x + 6} - 3 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20-%205%20e%5E%7Bx%20%2B%206%7D%20-%203%20" alt="LaTeX: \displaystyle f(x) = - 5 e^{x + 6} - 3 " data-equation-content=" \displaystyle f(x) = - 5 e^{x + 6} - 3 " /> . 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 left 6 units and down 3 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 5. The horizontal asymptote is <img class="equation_image" title=" \displaystyle y= -3 " src="/equation_images/%20%5Cdisplaystyle%20y%3D%20-3%20" alt="LaTeX: \displaystyle y= -3 " data-equation-content=" \displaystyle y= -3 " /> . The range is <img class="equation_image" title=" \displaystyle (-\infty, -3) " src="/equation_images/%20%5Cdisplaystyle%20%28-%5Cinfty%2C%20-3%29%20" alt="LaTeX: \displaystyle (-\infty, -3) " data-equation-content=" \displaystyle (-\infty, -3) " /> <br><?xml version="1.0" encoding="UTF-8"?>
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