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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 102b
Chapter 3, Problem 102b

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.
Graph of function ƒ with points marked, illustrating transformation y = ƒ(x-2).
(b) y = ƒ(x-2)

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Identify the transformation represented by the function y = ƒ(x - 2). This is a horizontal shift of the original function ƒ(x) to the right by 2 units.
Take each key point on the original graph of ƒ(x) and shift its x-coordinate 2 units to the right, while keeping the y-coordinate the same.
For example, the point (-8, 0) on ƒ(x) will move to (-8 + 2, 0) = (-6, 0) on y = ƒ(x - 2).
Similarly, the point (-4, 8) will move to (-4 + 2, 8) = (-2, 8), the point (0, 0) will move to (2, 0), and the point (16, 0) will move to (18, 0).
After shifting all points, sketch the new graph by connecting these transformed points with the same shape and curvature as the original graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Transformation - Horizontal Shifts

A horizontal shift of a function occurs when the input variable x is replaced by (x - h), where h is a constant. This shifts the graph of the function h units to the right if h is positive, and h units to the left if h is negative. For example, y = f(x - 2) shifts the graph of y = f(x) two units to the right.
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Shifts of Functions

Graphing Functions Using Key Points

To graph a transformed function, it is helpful to identify and shift key points from the original graph. Each point (x, y) on the original graph moves to (x + h, y) for a horizontal shift by h units. Plotting these new points and connecting them helps visualize the transformed function accurately.
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Graphing Rational Functions Using Transformations

Interpreting Piecewise or Composite Graphs

When a function graph consists of multiple segments or shapes, each segment must be transformed consistently. Understanding how each part behaves and shifts ensures the entire graph is correctly redrawn after transformation, preserving the shape and relative positions of all segments.
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Function Composition
Related Practice
Textbook Question

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.

(c) y = ƒ(x+3) - 2

Textbook Question

Let ƒ(x)=3x2−4ƒ(x) = 3x^2 - 4 and g(x)=x2−3x−4g(x) = x^2 - 3x -4. Find each of the following.

(f+g)(2k)(f+g)(2k)

Textbook Question

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.

(a) y = ƒ(x) +3

Textbook Question

Each of the following graphs is obtained from the graph of ƒ(x)=|x| or g(x)=√x by applying several of the transformations discussed in this section. Describe the transformations and give an equation for the graph.

Textbook Question

Let ƒ(x) = 3x -4. Find an equation for each reflection of the graph of ƒ(x). across the y-axis

Textbook Question

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.

(d) y = |ƒ(x)|