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

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 75

6장, 문제 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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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.