Consider the following nonlinear system. Work Exercises 75 –80 in order. y = | x - 1 | y = x2 - 4 How is the graph of y = | x - 1 | obtained by transforming the graph of y = | x |?

Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 75

Chapter 6, Problem 75

Consider the following nonlinear system. Work Exercises 75 –80 in order.
y = | x - 1 |
y = x² - 4
How is the graph of y = | x - 1 | obtained by transforming the graph of y = | x |?

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Recall the parent function for the absolute value is $y = |x|$, which has a V-shape with its vertex at the origin $(0,0)$.

The function $y = |x - 1|$ represents a horizontal shift of the parent function $y = |x|$.

Specifically, the expression inside the absolute value, $x - 1$, indicates a shift to the right by 1 unit because replacing $x$ with $x - h$ shifts the graph $h$ units to the right.

Therefore, the vertex of the graph of $y = |x - 1|$ moves from $(0,0)$ to $(1,0)$, maintaining the same V-shape but shifted horizontally.

In summary, the graph of $y = |x - 1|$ is obtained by taking the graph of $y = |x|$ and shifting it 1 unit to the right along 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 Function and Its Graph

The absolute value function y = |x| produces a V-shaped graph with its vertex at the origin (0,0). It outputs the distance of x from zero, making all values non-negative. Understanding this basic shape is essential before analyzing transformations.

Horizontal Shifts of Functions

A horizontal shift moves the graph left or right without changing its shape. For y = |x - 1|, the graph of y = |x| shifts 1 unit to the right because subtracting 1 inside the function moves the vertex from (0,0) to (1,0).

Graphical Interpretation of Function Transformations

Transformations like shifts, stretches, and reflections alter a graph's position or shape. Recognizing how changes inside the function's argument affect the graph helps in visualizing and sketching the new function accurately.

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Consider the following nonlinear system. Work Exercises 75 –80 in order.

y = | x - 1 |

y = x² - 4

Use the definition of absolute value to write y = | x - 1 | as a piecewise-defined function.

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Textbook Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

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