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

Problem

ƒ(x)=−2*(log_1/3)(x−1)−1

Solution

1. Identify the parent function as y=(log_1/3)(x)

2. Determine the horizontal shift by looking at the argument x−1 which moves the graph 1 unit to the right.

3. Identify the vertical asymptote by setting the argument to zero, resulting in x=1

4. Apply the vertical stretch and reflection by multiplying the yvalues of the parent function by −2

5. Determine the vertical shift by subtracting 1 from the yvalues, moving the graph down 1 unit.

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

ƒ(2)=−2*(log_1/3)(2−1)−1

ƒ(2)=−2*(log_1/3)(1)−1

ƒ(2)=−2*(0)−1=−1

7. Calculate another point. For x=4

ƒ(4)=−2*(log_1/3)(4−1)−1

ƒ(4)=−2*(log_1/3)(3)−1

ƒ(4)=−2*(−1)−1=1

Final Answer

ƒ(x)=−2*(log_1/3)(x−1)−1

The graph is a logarithmic curve with a vertical asymptote at x=1 passing through points (2,−1) and (4,1)

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