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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 95
Chapter 3, Problem 95

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.
Graph of an absolute value function transformed with vertex at (1,4) on a coordinate plane.

Verified step by step guidance
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Step 1: Identify the base function. The graph resembles the shape of the absolute value function \(f(x) = |x|\), which has a 'V' shape with its vertex at the origin (0,0).
Step 2: Observe the vertex shift. The vertex of the graph is at \((-3, 5)\) instead of at the origin, indicating a horizontal shift 3 units to the left and a vertical shift 5 units up.
Step 3: Analyze the slopes of the lines. The left side of the graph has a positive slope of 1, and the right side has a negative slope of -2, which means the right side is steeper than the left side. This suggests a vertical stretch by a factor of 2 on the right side and no stretch on the left side.
Step 4: Write the piecewise function. Since the graph is based on \(|x|\), which is defined as \(f(x) = -x\) for \(x < 0\) and \(f(x) = x\) for \(x \geq 0\), apply the transformations:
For \(x < -3\), the function is \(y = (x + 3) + 5\) (shifted left and up). For \(x \geq -3\), the function is \(y = -2(x + 3) + 5\) (shifted left and up, and vertically stretched by 2 with reflection).

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Key Concepts

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

Absolute Value Function

The absolute value function, f(x) = |x|, produces a V-shaped graph with its vertex at the origin. It outputs the distance of x from zero, always non-negative. Understanding its basic shape helps in identifying transformations like shifts, reflections, and stretches.
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Function Composition

Transformations of Functions

Transformations include shifts (horizontal and vertical), reflections (across axes), stretches, and compressions. These changes alter the graph's position, orientation, or size. Recognizing how each transformation affects the graph is essential for writing the new equation.
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Domain & Range of Transformed Functions

Piecewise Linear Functions

A piecewise linear function is defined by different linear expressions over intervals. The graph in the image shows two linear segments joined at a vertex, indicating a transformation of the absolute value function. Understanding piecewise definitions helps in expressing the transformed function accurately.
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Linear Inequalities
Related Practice
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

What is the relationship between the graphs of ƒ(x)=|x| and g(x)=|-x|?

Textbook Question

Describe how the graph of each function can be obtained from the graph of ƒ(x) = |x|. g(x) = -|x|

Textbook Question

For each function graphed, give the minimum and maximum values of ƒ(x) and the x-values at which they occur.

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Textbook Question

Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. |x| = |y|

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Textbook Question

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