Graph f(x)=2 log base 3 of x-1-3

Problem

ƒ(x)=2*(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 the points (1,0) and (3,1)

2. Determine the horizontal shift from the argument (x−1) Setting x−1=0 gives the new vertical asymptote at x=1 This shifts the entire graph 1 unit to the right.

3. Apply the vertical stretch by the factor of 2 The yvalues of the parent function are multiplied by 2 making the graph steeper.

4. Determine the vertical shift from the constant −3 This shifts the entire graph down 3 units.

5. Calculate key points to plot the graph.
   For x=2

ƒ(2)=2*(log_3)(2−1)−3

ƒ(2)=2*(0)−3=−3

Point: (2,−3)
For x=4

ƒ(4)=2*(log_3)(4−1)−3

ƒ(4)=2*(1)−3=−1

Point: (4,−1)
For x=10

ƒ(10)=2*(log_3)(10−1)−3

ƒ(10)=2*(2)−3=1

Point: (10,1)

6. Sketch the curve starting near the vertical asymptote x=1 passing through the calculated points, and increasing slowly as x increases.

Final Answer

ƒ(x)=2*(log_3)(x−1)−3

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