Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Explore thousands of questions and answers from knowledgeable experts in various fields on our Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly 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).
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.