Graph f(x)=- log base 3 of x-1+3

Problem

ƒ(x)=−(log_3)(x−1)+3

Solution

1. Identify the parent function and its key characteristics. The base function is y=(log_3)(x) which has a vertical asymptote at x=0 and passes through (1,0) and (3,1)

2. Determine the horizontal shift by looking at the argument x−1 Setting x−1=0 gives x=1 which is the new vertical asymptote. The graph shifts right by 1 unit.

3. Apply the reflection across the x-axis due to the negative sign in front of the logarithm. This changes the y-values from y to −y causing the graph to decrease as x increases.

4. Determine the vertical shift by looking at the constant +3 The entire graph shifts up by 3 units.

5. Calculate key points to plot the graph.
   For x=2 ƒ(2)=−(log_3)(2−1)+3=−(log_3)(1)+3=0+3=3 Point: (2,3)
   For x=4 ƒ(4)=−(log_3)(4−1)+3=−(log_3)(3)+3=−1+3=2 Point: (4,2)
   For x=10 ƒ(10)=−(log_3)(10−1)+3=−(log_3)(9)+3=−2+3=1 Point: (10,1)

6. Sketch the curve starting near the vertical asymptote x=1 passing through the calculated points, and continuing to decrease slowly as x approaches infinity.

Final Answer

ƒ(x)=−(log_3)(x−1)+3

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