Question

which equation is represented by the graph below?

graph showing a curve starting at (0,2) and increasing slowly as x increases, passing through (1,3)

• (y = e^x + 5)
• (y = e^x + 4)
• (y = \ln x + 4)

Explanation:

⚡ Using what you learned: Understanding Logarithmic Functions

Step 1: Identify key features of the graph

Observe the behavior and key points on the given graph:

• The graph has a vertical asymptote at the y-axis, \(x = 0\).
• The domain is \(x > 0\).
• The graph passes through the point \((1, 4)\).
• The graph passes through another clear point near \((2.7, 5)\), which corresponds to \((e, 5)\).

These features (vertical asymptote at \(x = 0\) and domain \(x > 0\)) are characteristic of a logarithmic function, \(y = \ln(x) + c\), rather than an exponential function, \(y = e^x + c\).

Step 2: Test the logarithmic option

Test the point \((1, 4)\) in the logarithmic equation option:

$$y = \ln(x) + 4$$

Substitute \(x = 1\):

$$y = \ln(1) + 4$$

$$y = 0 + 4 = 4$$

This matches the point \((1, 4)\) on the graph perfectly.

Answer:

\(y = \ln(x) + 4\)