describe the transformation of y=log (x-4) -1
2
from the graph of y=log X.
2
then state the asymptote and the domain range.
To describe the transformation of y=log((x-4)^2)-1 from the graph of y=log(x^2), we can break it down into three components: the horizontal shift, the vertical shift, and the vertical compression.
1. Horizontal Shift:
The function y=log(x^2) has its vertex (the lowest point on the graph) at x=0. By subtracting 4 inside the logarithm function, the vertex of y=log((x-4)^2) will shift to x=4, resulting in a horizontal shift of +4 units to the right.
2. Vertical Shift:
The original function y=log(x^2) has its horizontal asymptote at y=0, and its graph is symmetric about the y-axis. By subtracting 1 after squaring in the logarithm function, the entire graph of y=log((x-4)^2)-1 will shift down by 1 unit vertically.
3. Vertical Compression:
The exponent 2 inside the logarithm function in y=log((x-4)^2)-1 indicates a vertical compression by a factor of 1/2. This means that the graph will be compressed vertically compared to y=log(x^2), making it steeper.
As for the asymptote and domain range:
- The asymptote of the original function y=log(x^2) is y=0.
- The asymptote of the transformed function y=log((x-4)^2)-1 will still be y=0, as both the horizontal shift and vertical shift do not affect the asymptote.
- The domain of the original function y=log(x^2) is all real numbers except x=0 (since the logarithm of zero is undefined).
- The domain of the transformed function y=log((x-4)^2)-1 will be all real numbers except x=4 (due to the horizontal shift).
So, the asymptote of y=log((x-4)^2)-1 is y=0, and the domain range is all real numbers except x=4.
to get from y = logx to y = log(x-4) - 1
translate the original 4 units to the right, then down 1 unit
domain of new function: x > 4
range: all values of y
the line x = 4 is a vertical asymptote, no other asymptotes.