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**Transforming Functions: Understanding Shifts and Reflections** In this exercise, we aim to describe how to transform the function \( f(x) = \sqrt{x} \) into the graph of \( f(x) = -\sqrt{x} + 6 \). Below are the options provided for this transformation: - **Option 1:** Shift the graph left 6 units and then reflect across the x-axis. - **Option 2:** Shift the graph left 6 units and then reflect across the y-axis. - **Option 3:** Shift the graph right 6 units and then reflect across the x-axis. - **Option 4:** Shift the graph up 6 units and then reflect across the y-axis. Transforming the given function involves a series of steps that include shifting and reflecting the graph. ### Step-by-Step Transformation 1. **Reflection Across the x-axis:** - To reflect \( \sqrt{x} \) across the x-axis, we multiply the function by -1, resulting in \( -\sqrt{x} \). 2. **Vertical Shift:** - To shift the graph of \( -\sqrt{x} \) up by 6 units, we add 6 to the function, resulting in \( -\sqrt{x} + 6 \). ### Summary The correct transformation sequence is: - Reflect \( \sqrt{x} \) across the x-axis to get \( -\sqrt{x} \). - Shift the resulting graph up by 6 units to get \( -\sqrt{x} + 6 \). Hence, the correct option is: - **Option 4:** Shift the graph up 6 units and then reflect across the y-axis. (Note: The correct process described is reflection across the x-axis, not y-axis, suggesting the options provided might contain a typo. Clarify the concept focus is the correct process described here). This explanation helps learners understand the individual steps of transforming a basic function to help grasp the fundamental concepts of shifts and reflections in function graphs.

**Transforming Functions: Understanding Shifts and Reflections** In this exercise, we aim to describe how to transform the function \( f(x) = \sqrt{x} \) into the graph of \( f(x) = -\sqrt{x} + 6 \). Below are the options provided for this transformation: - **Option 1:** Shift the graph left 6 units and then reflect across the x-axis. - **Option 2:** Shift the graph left 6 units and then reflect across the y-axis. - **Option 3:** Shift the graph right 6 units and then reflect across the x-axis. - **Option 4:** Shift the graph up 6 units and then reflect across the y-axis. Transforming the given function involves a series of steps that include shifting and reflecting the graph. ### Step-by-Step Transformation 1. **Reflection Across the x-axis:** - To reflect \( \sqrt{x} \) across the x-axis, we multiply the function by -1, resulting in \( -\sqrt{x} \). 2. **Vertical Shift:** - To shift the graph of \( -\sqrt{x} \) up by 6 units, we add 6 to the function, resulting in \( -\sqrt{x} + 6 \). ### Summary The correct transformation sequence is: - Reflect \( \sqrt{x} \) across the x-axis to get \( -\sqrt{x} \). - Shift the resulting graph up by 6 units to get \( -\sqrt{x} + 6 \). Hence, the correct option is: - **Option 4:** Shift the graph up 6 units and then reflect across the y-axis. (Note: The correct process described is reflection across the x-axis, not y-axis, suggesting the options provided might contain a typo. Clarify the concept focus is the correct process described here). This explanation helps learners understand the individual steps of transforming a basic function to help grasp the fundamental concepts of shifts and reflections in function graphs.

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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**Transforming Functions: Understanding Shifts and Reflections** In this exercise, we aim to describe how to transform the function \( f(x) = \sqrt{x} \) into the graph of \( f(x) = -\sqrt{x} + 6 \). Below are the options provided for this transformation: - **Option 1:** Shift the graph left 6 units and then reflect across the x-axis. - **Option 2:** Shift the graph left 6 units and then reflect across the y-axis. - **Option 3:** Shift the graph right 6 units and then reflect across the x-axis. - **Option 4:** Shift the graph up 6 units and then reflect across the y-axis. Transforming the given function involves a series of steps that include shifting and reflecting the graph. ### Step-by-Step Transformation 1. **Reflection Across the x-axis:** - To reflect \( \sqrt{x} \) across the x-axis, we multiply the function by -1, resulting in \( -\sqrt{x} \). 2. **Vertical Shift:** - To shift the graph of \( -\sqrt{x} \) up by 6 units, we add 6 to the function, resulting in \( -\sqrt{x} + 6 \). ### Summary The correct transformation sequence is: - Reflect \( \sqrt{x} \) across the x-axis to get \( -\sqrt{x} \). - Shift the resulting graph up by 6 units to get \( -\sqrt{x} + 6 \). Hence, the correct option is: - **Option 4:** Shift the graph up 6 units and then reflect across the y-axis. (Note: The correct process described is reflection across the x-axis, not y-axis, suggesting the options provided might contain a typo. Clarify the concept focus is the correct process described here). This explanation helps learners understand the individual steps of transforming a basic function to help grasp the fundamental concepts of shifts and reflections in function graphs.
Transcribed Image Text:**Transforming Functions: Understanding Shifts and Reflections** In this exercise, we aim to describe how to transform the function \( f(x) = \sqrt{x} \) into the graph of \( f(x) = -\sqrt{x} + 6 \). Below are the options provided for this transformation: - **Option 1:** Shift the graph left 6 units and then reflect across the x-axis. - **Option 2:** Shift the graph left 6 units and then reflect across the y-axis. - **Option 3:** Shift the graph right 6 units and then reflect across the x-axis. - **Option 4:** Shift the graph up 6 units and then reflect across the y-axis. Transforming the given function involves a series of steps that include shifting and reflecting the graph. ### Step-by-Step Transformation 1. **Reflection Across the x-axis:** - To reflect \( \sqrt{x} \) across the x-axis, we multiply the function by -1, resulting in \( -\sqrt{x} \). 2. **Vertical Shift:** - To shift the graph of \( -\sqrt{x} \) up by 6 units, we add 6 to the function, resulting in \( -\sqrt{x} + 6 \). ### Summary The correct transformation sequence is: - Reflect \( \sqrt{x} \) across the x-axis to get \( -\sqrt{x} \). - Shift the resulting graph up by 6 units to get \( -\sqrt{x} + 6 \). Hence, the correct option is: - **Option 4:** Shift the graph up 6 units and then reflect across the y-axis. (Note: The correct process described is reflection across the x-axis, not y-axis, suggesting the options provided might contain a typo. Clarify the concept focus is the correct process described here). This explanation helps learners understand the individual steps of transforming a basic function to help grasp the fundamental concepts of shifts and reflections in function graphs.
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