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What transformation f(x) = 2-2 (red) to g (x) (blue)? is shown in the image from g(x) = O Horizontal Shift of 4 units O Vertical Shift of 4 units O Horizontal Stretch by a factor of 4 O Vertical Stretch by a fact of 4

What transformation f(x) = 2-2 (red) to g (x) (blue)? is shown in the image from g(x) = O Horizontal Shift of 4 units O Vertical Shift of 4 units O Horizontal Stretch by a factor of 4 O Vertical Stretch by a fact of 4

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Transformation of Functions: Graph Analysis #### What transformation is shown in the image from \( f(x) = 2^x - 2 \) (red) to \( g(x) \) (blue)? In the provided graph, we observe two functions: - The **red curve** represents the function \( f(x) = 2^x - 2 \). - The **blue curve** represents the function \( g(x) \), whose equation we need to determine based on its transformation from \( f(x) \). The graph illustrates the behavior and relationship between these two functions plotted on a Cartesian coordinate system with marked axes and grid lines for clarity. ### Visual Analysis of the Graph 1. **Red Curve ( \( f(x) = 2^x - 2 \) )**: - This function is an exponential function shifted down by 2 units. - Characteristics: - Asymptote: Horizontal asymptote at \( y = -2 \). - Intercept: Crosses the y-axis at \( y = -1 \). 2. **Blue Curve ( \( g(x) \) )**: - Appears to be a transformed version of the red curve. - Key Observation: The blue curve is linear (straight line), suggesting a transformation involving linearization. - The intercept at \( x = 0 \) which appears at \( y = 0 \), indicating no vertical shift for this particular point. ### Determining Transformation To deduce the specific transformation, recognize the following: - **Exponential to Linear**: If the function transformed from an exponential to a linear function, it is likely a horizontal stretch or vertical stretch. From the provided transformation options: 1. **Horizontal Shift of 4 units**: This would result in a translation along the x-axis. 2. **Vertical Shift of 4 units**: This would result in a translation along the y-axis. 3. **Horizontal Stretch by a factor of 4**: This would widen the graph horizontally. 4. **Vertical Stretch by a factor of 4**: This would stretch the graph vertically, expanding its y-values by a factor. Given that we observe a change from an exponential function (non-linear) to a linear function, the appropriate transformation appears to be a **Vertical Stretch** by a factor of 4. ### Conclusion The transformation shown in the image from \( f(x) = 2
Transcribed Image Text:### Transformation of Functions: Graph Analysis #### What transformation is shown in the image from \( f(x) = 2^x - 2 \) (red) to \( g(x) \) (blue)? In the provided graph, we observe two functions: - The **red curve** represents the function \( f(x) = 2^x - 2 \). - The **blue curve** represents the function \( g(x) \), whose equation we need to determine based on its transformation from \( f(x) \). The graph illustrates the behavior and relationship between these two functions plotted on a Cartesian coordinate system with marked axes and grid lines for clarity. ### Visual Analysis of the Graph 1. **Red Curve ( \( f(x) = 2^x - 2 \) )**: - This function is an exponential function shifted down by 2 units. - Characteristics: - Asymptote: Horizontal asymptote at \( y = -2 \). - Intercept: Crosses the y-axis at \( y = -1 \). 2. **Blue Curve ( \( g(x) \) )**: - Appears to be a transformed version of the red curve. - Key Observation: The blue curve is linear (straight line), suggesting a transformation involving linearization. - The intercept at \( x = 0 \) which appears at \( y = 0 \), indicating no vertical shift for this particular point. ### Determining Transformation To deduce the specific transformation, recognize the following: - **Exponential to Linear**: If the function transformed from an exponential to a linear function, it is likely a horizontal stretch or vertical stretch. From the provided transformation options: 1. **Horizontal Shift of 4 units**: This would result in a translation along the x-axis. 2. **Vertical Shift of 4 units**: This would result in a translation along the y-axis. 3. **Horizontal Stretch by a factor of 4**: This would widen the graph horizontally. 4. **Vertical Stretch by a factor of 4**: This would stretch the graph vertically, expanding its y-values by a factor. Given that we observe a change from an exponential function (non-linear) to a linear function, the appropriate transformation appears to be a **Vertical Stretch** by a factor of 4. ### Conclusion The transformation shown in the image from \( f(x) = 2
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