Graph - log base 3 of x-1+3

Problem

y=−(log_3)(x−1)+3

Solution

1. Identify the parent function as y=(log_3)(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 x−1 which moves the graph and the vertical asymptote 1 unit to the right. The new vertical asymptote is x=1

3. Apply the vertical reflection caused by the negative sign in front of the logarithm, which flips the graph across the horizontal axis.

4. Determine the vertical shift by adding 3, which moves the entire graph up 3 units.

5. Calculate key points to plot the curve. For x=2 y=−(log_3)(2−1)+3=3 For x=4 y=−(log_3)(4−1)+3=−(log_3)(3)+3=−1+3=2

Final Answer

y=−(log_3)(x−1)+3

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