Graph - natural log of x-1+3

Problem

y=−ln(x−1)+3

Solution

1. Identify the parent function and its basic properties. The parent function is ƒ(x)=ln(x) which has a vertical asymptote at x=0 and passes through the point (1,0)

2. Determine the horizontal shift by looking at the argument of the logarithm. The term (x−1) indicates a shift to the right by 1 unit. The new vertical asymptote is x=1

3. Apply the reflection across the x-axis. The negative sign in front of the natural log, −ln(x−1) reflects the graph vertically, meaning the function now decreases as x increases.

4. Determine the vertical shift by looking at the constant term. The +3 indicates a shift upward by 3 units.

5. Find key points to plot the graph. When x=2 y=−ln(2−1)+3=−ln(1)+3=0+3=3 So, the point (2,3) is on the graph. When x=e+1 y=−ln(e)+3=−1+3=2 So, the point (e+1,2)≈(3.72,2) is on the graph.

Final Answer

y=−ln(x−1)+3

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