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Chapter 9: Problem 68

Consider the following nonlinear system. $$ \begin{array}{l} y=|x-1| \\ y=x^{2}-4 \end{array} $$ Work Exercise, to see how concepts from previous chapters are related to the graphs and the solutions of this system. How is the graph of \(y=|x-1|\) obtained by transforming the graph of \(y=|x| ?\)

Short Answer

Expert verified
Shift the graph of \(y = |x|\) right by 1 unit to obtain \(y = |x-1|\).

Step by step solution

01

Understand the basic graph

The function you need to understand first is the absolute value function. The graph of the function \(y = |x|\) is a V-shaped curve with its vertex at the origin (0, 0), opening upwards.
02

Horizontal Shift

The graph of \(y = |x-1|\) is obtained by shifting the graph of \(y = |x|\) horizontally to the right by 1 unit. This is because the expression inside the absolute value is \(x - 1\), which means every x-coordinate is adjusted by adding 1.
03

Plot the new graph

Transform the original graph by moving each point on \(y = |x|\) to their new positions. For example, the vertex at (0,0) of \(y = |x|\) will be moved to (1,0) for the graph of \(y = |x-1|\).
04

Final Graph

The final graph of \(y = |x-1|\) is a V-shaped curve with the vertex at (1, 0), opening upwards, indicating the transformation from the original \(y = |x|\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

absolute value function
The absolute value function is an essential nonlinear function. Its general form is given by the equation \(|x|\), where \(|x|\) represents the distance of x from zero on the real number line.
In simple terms, it makes any negative number positive.
Graphically, the absolute value function \(y = |x|\) is a V-shaped curve.
Its vertex is at the origin (0, 0) and it opens upwards, reflecting its non-negative value aspect.
This basic understanding is crucial before diving into transformations.
Let's break down specific transformations one by one.
graph transformations
Graph transformations are functions applied to a graph to change its position, shape, or orientation.
They are very useful in understanding how basic functions are manipulated in more complex scenarios.
For instance, in the function \(y = |x|\), we can apply transformations to shift the graph or change its scale.
  • Horizontal Shifts
  • Vertical Shifts
  • Reflections
  • Stretching/Shrinking
Each of these transformations follows specific rules and affects the graph in predictable ways.
In this context, we will focus more on horizontal shifts to understand the behavior of \(y = |x-1|\).
horizontal shift
A horizontal shift moves the graph left or right across the Cartesian plane.
To achieve a horizontal shift, we modify the function inside the absolute value.
For example, \(|x-h|\) shifts the graph horizontally by h units:
  • If h > 0, the graph shifts right
  • If h < 0, the graph shifts left
In the exercise, the function is \(y = |x-1|\).
Since the value subtracted from x is positive (1), the graph shifts 1 unit to the right.
Every point of the base graph \(y = |x|\) is adjusted accordingly:
  • The origin vertex (0,0) moves to (1,0)
  • Any other x-coordinate shifts right by 1 unit
This fundamental shift assists in plotting the transformed graph accurately.
vertex transformation
The vertex of the absolute value function is its lowest point since the graph opens upward.
It's crucial because it indicates the point of symmetry for the function.
For the basic graph \(y = |x|\), the vertex is at (0,0).
However, when applying transformations, especially horizontal shifts, the vertex moves accordingly.
In our function \(y = |x-1|\), we've determined the graph shifts 1 unit to the right.
This moves the vertex from (0,0) to (1,0).
Other points on the old graph adjust similarly, but the vertex is the most notable shift.
By observing the vertex transformation, you gain better insight into how the entire graph is altered during transformations.

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Most popular questions from this chapter

Find the equation of the parabola \(y=a x^{2}+b x+c\) that passes through the points \((2,3),(-1,0),\) and \((-2,2)\) Find the equation of the parabola \(y=a x^{2}+b x+c\) that passes through the points \((-2,4),(2,2),\) and \((4,9)\)

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