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Find the function that is finally graphed after the following transformations are applied to the graph of y = x in the order listed. (1) Shift up 3 units (2) Reflect about the x-axis (3) Reflect about the y-axis

Find the function that is finally graphed after the following transformations are applied to the graph of y = x in the order listed. (1) Shift up 3 units (2) Reflect about the x-axis (3) Reflect about the y-axis

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Title: Exploring Function Transformations Content: Find the function that is finally graphed after the following transformations are applied to the graph of \( y = \sqrt{x} \) in the order listed. 1. Shift up 3 units 2. Reflect about the x-axis 3. Reflect about the y-axis --- **Explanation of Transformations:** 1. **Shift Up 3 Units:** - The transformation will change the function to \( y = \sqrt{x} + 3 \). This moves the graph vertically upwards by 3 units. 2. **Reflect About the x-axis:** - The reflection will alter the function to \( y = -(\sqrt{x} + 3) \), which means the graph is flipped over the x-axis. 3. **Reflect About the y-axis:** - This further changes the function to \( y = -\sqrt{-x} - 3 \). The graph is now flipped over the y-axis as well. **Resulting Function:** - After applying all transformations, the final function is \( y = -\sqrt{-x} - 3 \). This sequence of transformations demonstrates how basic algebraic manipulations can significantly shift and reflect a function on a graph, offering insights into function behavior and symmetry.
Transcribed Image Text:Title: Exploring Function Transformations Content: Find the function that is finally graphed after the following transformations are applied to the graph of \( y = \sqrt{x} \) in the order listed. 1. Shift up 3 units 2. Reflect about the x-axis 3. Reflect about the y-axis --- **Explanation of Transformations:** 1. **Shift Up 3 Units:** - The transformation will change the function to \( y = \sqrt{x} + 3 \). This moves the graph vertically upwards by 3 units. 2. **Reflect About the x-axis:** - The reflection will alter the function to \( y = -(\sqrt{x} + 3) \), which means the graph is flipped over the x-axis. 3. **Reflect About the y-axis:** - This further changes the function to \( y = -\sqrt{-x} - 3 \). The graph is now flipped over the y-axis as well. **Resulting Function:** - After applying all transformations, the final function is \( y = -\sqrt{-x} - 3 \). This sequence of transformations demonstrates how basic algebraic manipulations can significantly shift and reflect a function on a graph, offering insights into function behavior and symmetry.
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