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Describe how the graph of y=2log3(x−4)+2
compares to the parent graph. Discuss any changes in the asymptotes, domain, and range
To compare the graph of y = 2log3(x-4)+2 to the parent graph of y = log(x), we first need to identify the transformations that have been applied to the parent graph.
The parent graph of y = log(x) has a vertical asymptote at x = 0 and passes through the point (1,0). Its domain is x > 0 and its range is all real numbers.
For the given function y = 2log3(x-4)+2, the transformations are as follows:
1. The graph has been shifted 4 units to the right (in the positive x-direction) due to the term -4 inside the log function.
2. The graph has been vertically stretched by a factor of 2 due to the coefficient 2 in front of the log function.
3. The entire graph has been shifted 2 units upwards due to the +2 outside the log function.
As a result of these transformations:
1. The vertical asymptote of the function will be x = 4, instead of x = 0 as in the parent graph.
2. The domain of the function will be x > 4, since the shift to the right moves all values of x by 4 units.
3. The range of the function will be all real numbers, since the vertical stretch does not affect the y-values.
Therefore, the graph of y = 2log3(x-4)+2 will be similar to the parent graph of y = log(x) but shifted to the right, vertically stretched, and shifted upwards.