To identify the type of transformation and describe the change from the function [tex]\( f(x) = \log x \)[/tex] to [tex]\( g(x) = \log(x) - 3 \)[/tex], let's break down the steps in detail:
1. Understand the Original Function:
The original function is [tex]\( f(x) = \log x \)[/tex]. This is a logarithmic function, which typically has its graph passing through the point (1,0) and increases slowly as [tex]\( x \)[/tex] increases.
2. Identify the Transformed Function:
The transformed function is [tex]\( g(x) = \log(x) - 3 \)[/tex]. This function is derived from [tex]\( f(x) \)[/tex] by subtracting 3 from [tex]\( \log x \)[/tex].
3. Determine the Type of Transformation:
When a constant is subtracted from the entire function [tex]\( f(x) \)[/tex], it results in a vertical shift. Specifically, subtracting 3 means that every point on the graph of [tex]\( f(x) \)[/tex] is moved down by 3 units.
4. Describe the Change:
For the given transformation, [tex]\( g(x) = \log(x) - 3 \)[/tex] indicates that the graph of [tex]\( f(x) = \log x \)[/tex] is shifted vertically downward by 3 units. Thus, every [tex]\( y \)[/tex]-value of the original function [tex]\( f(x) \)[/tex] is decreased by 3.
Conclusion:
The transformation from [tex]\( f(x) = \log x \)[/tex] to [tex]\( g(x) = \log(x) - 3 \)[/tex] is a vertical shift. Specifically, the graph of [tex]\( f(x) \)[/tex] is shifted 3 units downward to produce [tex]\( g(x) \)[/tex].
So, the type of transformation is:
- Vertical shift
And the description of the change is:
- The graph of [tex]\( f(x) = \log x \)[/tex] is shifted 3 units downward to get [tex]\( g(x) = \log(x) - 3 \)[/tex].
This completes the detailed explanation of identifying and describing the transformation.
