**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 FormIdentify 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 ShiftRewrite the function incorporating the shift.### Step 4: Validate with an ExampleConsider 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.

We have to determine the process of horizontal shift.

Answered: Explain the process of correctly…

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

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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Question

**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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