Question

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

Explanation:

⚡ Using what you learned: Understanding Logarithmic Functions

Step 1: Identify key points of the parent function

The parent function is:

$$ y = \ln x $$

Key points for the natural logarithm function include:

• When \( x = 1 \), \( y = \ln(1) = 0 \), giving the point \((1, 0)\).
• When \( x = e \approx 2.718 \), \( y = \ln(e) = 1 \), giving the point \((2.718, 1)\).
• There is a vertical asymptote at \( x = 0 \).

Step 2: Apply the vertical shift

The given equation is:

$$ y = \ln x + 1 $$

This represents a vertical shift upward by \( 1 \) unit. We apply this shift to our key points:

• The point \((1, 0)\) shifts up to \((1, 1)\).
• The point \((e, 1) \approx (2.718, 1)\) shifts up to \((e, 2) \approx (2.718, 2)\).
• The vertical asymptote remains at \( x = 0 \) (the y-axis).

Step 3: Match with the correct graph

Let's examine the given graphs to find the one containing the point \((1, 1)\) with a vertical asymptote at \( x = 0 \):

• First Graph:
• At \( x = 1 \), the curve passes through \( y = 1 \).
• The vertical asymptote is at \( x = 0 \) (the y-axis).
• This matches our shifted function \( y = \ln x + 1 \).

• Second Graph:
• At \( x = 1 \), the curve passes through \( y = 0 \). This is the unshifted parent function \( y = \ln x \).

• Third Graph:
• The vertical asymptote is shifted to the right at \( x = 2 \), which would represent a horizontal shift like \( y = \ln(x - 2) \).

Answer:

The correct graph is the first graph (the leftmost option), which passes through the point \((1, 1)\) and has a vertical asymptote at the y-axis (\( x = 0 \)).