Question

which equation is represented by the graph below?
( y=ln x - 3 )
( y=ln x - 4 )
( y = e ^ { x } - 3 )
( y = e ^ { x } - 4 )

Explanation:

Step1: Analyze the general form of the function

The graph has a horizontal asymptote. For \(y = \ln x\), there is a vertical asymptote \(x = 0\), and it does not have a horizontal asymptote. For \(y=e^{x}\), the horizontal asymptote is \(y = 0\). After vertical - shift \(y = e^{x}+k\), the horizontal asymptote is \(y=k\). So the function is more likely an exponential function (since it has a horizontal asymptote), not a logarithmic function.

Step2: Use the \(y\) - intercept to determine the function

The \(y\) - intercept is the value of \(y\) when \(x = 0\).
For \(y=e^{x}-3\), when \(x = 0\), \(y=e^{0}-3=1 - 3=-2\) (incorrect).
For \(y=e^{x}-4\), when \(x = 0\), \(y=e^{0}-4=1 - 4=-3\) (correct).

Answer:

\(y = e^{x}-4\)