Question

find the equation of the asymptote for ( g(x)=e^{x - 5} ) using the graph. (type an equation.)

Explanation:

Step1: Recall the asymptote of the basic exponential function

The basic exponential function \(y = e^{x}\) has a horizontal asymptote \(y = 0\).

Step2: Consider the transformation

For the function \(g(x)=e^{x}-5\), it is a vertical translation of \(y = e^{x}\) down by 5 units.
When a function \(y = f(x)\) is translated down by \(k\) units (\(y=f(x)-k\)), the horizontal asymptote \(y = c\) of \(y = f(x)\) is translated to \(y=c - k\).
Since \(c = 0\) for \(y = e^{x}\) and \(k = 5\), the horizontal asymptote of \(g(x)=e^{x}-5\) is \(y=0 - 5\).

Answer:

\(y=-5\)