IDNLearn.com offers a unique blend of expert answers and community insights. Discover the reliable solutions you need with help from our comprehensive and accurate Q&A platform.
Sagot :
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).
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).
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com has the solutions you’re looking for. Thanks for visiting, and see you next time for more reliable information.