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

Problem

ƒ(x)=2*(log_1/3)(x−1)+3

Solution

1. Identify the parent function and its key characteristics. The base function is y=(log_1/3)(x) Since the base b=1/3 is between 0 and 1 the graph is decreasing. It has a vertical asymptote at x=0 and passes through (1,0) and (1/3,1)

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 multiplying the output by 2 This doubles the distance of each point from the horizontal axis, making the curve steeper.

4. Apply the vertical shift by adding 3 to the function. This moves the entire graph up 3 units.

5. Calculate key points to plot the graph accurately.

   • If x=2 ƒ(2)=2*(log_1/3)(2−1)+3=2*(0)+3=3 Point: (2,3)

   • If x=11/3=4/3 ƒ*(4/3)=2*(log_1/3)(1/3)+3=2*(1)+3=5 Point: (4/3,5)

   • If x=4 ƒ(4)=2*(log_1/3)(3)+3=2*(−1)+3=1 Point: (4,1)

6. Sketch the graph by drawing a smooth curve through the points (4/3,5) (2,3) and (4,1) approaching the vertical asymptote x=1 as x approaches 1 from the right.

Final Answer

ƒ(x)=2*(log_1/3)(x−1)+3

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