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Explain the process of correctly determining Horizontal Shifts. Include a mathematical example to support your explanation.

Explain the process of correctly determining Horizontal Shifts. Include a mathematical example to support your explanation.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Determining Horizontal Shifts in Graphs: A Step-by-Step Guide** Horizontal shifts occur when a graph shifts left or right on the coordinate plane. This type of transformation is crucial in many mathematical functions and can be understood with the following process: ### Step 1: Recognize the Standard Form Identify the standard form of the function. For any function \( f(x) \), the general form indicating a horizontal shift is \( f(x - h) \) where \( h \) is the shift value. ### Step 2: Determine the Direction of the Shift - If the expression within the function is \( (x - h) \), the graph shifts horizontally to the right by \( h \) units. - If the expression within the function is \( (x + h) \), the graph shifts horizontally to the left by \( h \) units. ### Step 3: Apply the Horizontal Shift Rewrite the function incorporating the shift. ### Step 4: Validate with an Example Consider the function \( f(x) = (x - 3)^2 \). - The term \( (x - 3) \) indicates a horizontal shift 3 units to the right. - The new function \( f(x) = (x - 3)^2 \) is the transformed function after the horizontal shift. **Graphical Representation:** Diagram: 1. Graph the standard function \( f(x) = x^2 \) which is a parabola opening upwards with its vertex at the origin (0,0). 2. Graph the shifted function \( f(x) = (x - 3)^2 \): - The vertex of the parabola shifts from (0,0) to (3,0). These steps help us accurately determine and visualize horizontal shifts in any given function, ensuring a clear understanding of transformations on the coordinate plane.
Transcribed Image Text:**Determining Horizontal Shifts in Graphs: A Step-by-Step Guide** Horizontal shifts occur when a graph shifts left or right on the coordinate plane. This type of transformation is crucial in many mathematical functions and can be understood with the following process: ### Step 1: Recognize the Standard Form Identify the standard form of the function. For any function \( f(x) \), the general form indicating a horizontal shift is \( f(x - h) \) where \( h \) is the shift value. ### Step 2: Determine the Direction of the Shift - If the expression within the function is \( (x - h) \), the graph shifts horizontally to the right by \( h \) units. - If the expression within the function is \( (x + h) \), the graph shifts horizontally to the left by \( h \) units. ### Step 3: Apply the Horizontal Shift Rewrite the function incorporating the shift. ### Step 4: Validate with an Example Consider the function \( f(x) = (x - 3)^2 \). - The term \( (x - 3) \) indicates a horizontal shift 3 units to the right. - The new function \( f(x) = (x - 3)^2 \) is the transformed function after the horizontal shift. **Graphical Representation:** Diagram: 1. Graph the standard function \( f(x) = x^2 \) which is a parabola opening upwards with its vertex at the origin (0,0). 2. Graph the shifted function \( f(x) = (x - 3)^2 \): - The vertex of the parabola shifts from (0,0) to (3,0). These steps help us accurately determine and visualize horizontal shifts in any given function, ensuring a clear understanding of transformations on the coordinate plane.
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