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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 104d
Chapter 3, Problem 104d

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; sketch y = |ƒ(x)| as per the question.
(d) y = |ƒ(x)|

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1
Understand that the function y = |ƒ(x)| means taking the absolute value of the original function's output. This means all y-values will be non-negative (y ≥ 0).
Identify the parts of the original graph where ƒ(x) is positive or zero. For these parts, the graph of y = |ƒ(x)| will remain the same because the absolute value of a positive number or zero is the number itself.
Locate the parts of the graph where ƒ(x) is negative. For these parts, reflect the graph above the x-axis by taking the positive value of the y-coordinates. In other words, change the negative y-values to their positive counterparts.
Specifically, for the segment from (0, 0) to (8, -4), reflect the points below the x-axis to above it. For example, the point (8, -4) will become (8, 4) in the graph of y = |ƒ(x)|.
Sketch the new graph by keeping the positive parts unchanged and reflecting the negative parts above the x-axis, ensuring the graph never goes below the x-axis.

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Key Concepts

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

Absolute Value of a Function

The absolute value of a function, denoted as |ƒ(x)|, transforms all negative output values of ƒ(x) into their positive counterparts. This means any part of the graph below the x-axis is reflected above it, while parts already above the x-axis remain unchanged.
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Function Composition

Graph Transformation

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For y = |ƒ(x)|, the transformation reflects the portion of the graph where ƒ(x) < 0 across the x-axis, effectively flipping negative y-values to positive.
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Intro to Transformations

Interpreting Key Points on a Graph

Key points such as intercepts and vertices help in accurately sketching transformed graphs. For y = |ƒ(x)|, points where ƒ(x) = 0 remain fixed, while points with negative y-values are reflected above the x-axis, aiding in precise graph construction.
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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

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)(−1)(f/g)(-1)

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

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

(b) y = ƒ(x-2)

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Textbook Question

Work each problem. Find a function g(x)=ax+b whose graph can be obtained by translating the graph of ƒ(x)=2x+5 up 2 units and to the left 3 units.

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