How to graph an exponential function - Algebra 1

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Question

What is the horizontal asymptote of the graph of the equation $y = -2e^{x-3} -5$ ?

Answer

The asymptote of this equation can be found by observing that $e^{x-3} > 0$ regardless of . We are thus solving for the value of as $e^{x-3}$ approaches zero.

$-2e^{x-3} < -2\cdot 0$

$-2e^{x-3} < 0$

$-2e^{x-3} -5 < 0-5$

$-2e^{x-3} -5 < -5$

So the value that cannot exceed is , and the line is the asymptote.

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Question

What is/are the asymptote(s) of the graph of the function

$f(x) = 5e^{x-2}-2$ ?

Answer

An exponential equation of the form $f(x) = ae^{x-h} +k$ has only one asymptote - a horizontal one at . In the given function, , so its one and only asymptote is .

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Question

$\textup{What is the \textit{x}-intercept of the equation } y=e^{x-1}-1?$

Answer

$\ \ \textup{Solve the equation } \0=e^{x-1} -1\ e^{x-1}=1\ {x-1}=\textup{ln}(1)=0\ x=1.\\textup{ Remember }e^0=1 \textup{, so when plugging } x=1 \textup{ in the initial equation, you obtain }\ e^{1-1}-1=e^0-1=1-1=0.$

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