To analyze the features of the function [tex]\( g(x) = -4 \log (x - 8) \)[/tex] in relation to the base function [tex]\( f(x) = \log x \)[/tex], we need to consider how various transformations affect the graph of the logarithmic function.
1. Horizontal Shift:
The term [tex]\((x - 8)\)[/tex] inside the logarithm indicates a horizontal shift. Specifically, the function [tex]\( g(x) \)[/tex] is shifted 8 units to the right compared to the base logarithmic function [tex]\( f(x) \)[/tex]. This means that instead of the argument being [tex]\( x \)[/tex], it is [tex]\( x - 8 \)[/tex].
- New vertical asymptote: The vertical asymptote of [tex]\( f(x) = \log x \)[/tex] is at [tex]\( x = 0 \)[/tex]. For [tex]\( g(x) = -4 \log (x - 8) \)[/tex], the vertical asymptote will move to [tex]\( x = 8 \)[/tex].
Therefore, the horizontal shift is 8 units to the right.
2. Vertical Scaling and Reflection:
The coefficient [tex]\(-4\)[/tex] outside the logarithm function affects both the reflection and the scaling of the function.
- Reflection over the x-axis: The negative sign indicates that the function is reflected across the x-axis. This means if [tex]\( f(x) \)[/tex] was positive, [tex]\( g(x) \)[/tex] will be negative and vice versa.
- Vertical Stretch: The factor 4 indicates that the function is stretched vertically by a factor of 4. This means that the values of [tex]\( g(x) \)[/tex] are four times further from the x-axis than they would be in the base function [tex]\( f(x) \)[/tex].
Therefore, the vertical transformation is a reflection over the x-axis, combined with a vertical stretch by a factor of 4.
To summarize, the two key features of the function [tex]\( g(x) = -4 \log (x - 8) \)[/tex] are:
- A horizontal shift of 8 units to the right.
- A vertical transformation that includes a reflection over the x-axis and a vertical stretch by a factor of 4.
Thus, the numerical representation of these transformations is:
- Horizontal shift: [tex]\( 8 \)[/tex] units to the right.
- Vertical scaling: [tex]\( -4 \)[/tex] (indicating both the reflection and stretching).
