Question

identify the graph of \\(y = \ln x + 1\\).

Explanation:

Analyze the parent function and its key points

Using the Logarithmic Function Transformations knowledge point

$$
LATEXBLOCK0
$$

Apply the vertical translation

Using the Logarithmic Function Transformations knowledge point

$$
LATEXBLOCK1
$$

Evaluate another point to confirm the graph

Using the Logarithmic Function Transformations knowledge point

$$
LATEXBLOCK2
$$

Match with the given options

Using the Logarithmic Function Transformations knowledge point

• First Graph (Left): Passes through \((1, 1)\) and has a vertical asymptote at \(x = -1\). This represents \(y = \ln(x+1) + 1\).
• Second Graph (Middle): Passes through \((1, 1)\) and has a vertical asymptote at \(x = 0\) (the y-axis). It also passes through approximately \((2.7, 2)\). This matches \(y = \ln x + 1\).
• Third Graph (Right): Passes through \((3, 1)\) and has a vertical asymptote at \(x = 2\). This represents \(y = \ln(x-2) + 1\).

Answer:

• (A) Graph with vertical asymptote at x = -1, passing through (1, 1)
• (B) Graph with vertical asymptote at x = 0, passing through (1, 1) (Correct answer)
• (C) Graph with vertical asymptote at x = 2, passing through (3, 1)