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Chapter 0: Problem 10

Sketch the graph of the equation by point plotting. $$y=|x|-1$$

Short Answer

Expert verified
The graph of the equation \(y = |x| - 1\) is a 'V' shape that is symmetric about the y-axis, shifted 1 unit down.

Step by step solution

01

Understand the Equation

The absolute value of a real number \(x\), denoted \(|x|\), is the non-negative value of \(x\). This essentially takes any input value and converts it into its non-negative equivalent. For the equation \(y = |x| - 1\), the '-1' subtracts 1 from whatever the absolute value of \(x\) is.
02

Compute Key Points

To plot an equation, a selection of key points are needed. In this case, some useful key points could be where \(x = -2, -1, 0, 1, 2\). As \(y\) is the absolute value \(x\) minus 1, we have the following key points: For \(x = -2, |x| = 2\), so \(y = 2 - 1 = 1\). For \(x = -1, |x| = 1\), so \(y = 1 - 1 = 0\). For \(x = 0, |x| = 0\), so \(y = 0 - 1 = -1\). For \(x = 1, |x| = 1\), so \(y = 1 - 1 = 0\). For \(x = 2, |x| = 2\), so \(y = 2 - 1 = 1\). These key points will be used to plot the graph.
03

Sketch the Graph

Plot the key points from step 2 on a graph. The coordinates to plot are (-2,1), (-1,0), (0,-1), (1,0), (2,1). The graph is a 'V' shape that is symmetric about the y-axis. Because of the '-1' term, this 'V' is shifted 1 unit down from what a normal \(|x|\) graph would look like.

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Key Concepts

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

Graph Sketching
When sketching the graph of an absolute value function, it's essential to start by understanding the basic shape. The graph of a simple absolute value function like \(y = |x|\) is a "V" shape that opens upwards. This occurs because absolute value outputs the non-negative value of any input. For our specific function, \(y = |x| - 1\), this "V" shape is shifted down by 1 unit.

To sketch the graph effectively, identify key points by choosing a range of x-values, computing corresponding y-values, and then plotting these on the graph. The symmetry of the absolute value function will aid in ensuring the graph is accurate.
Key Points
Key points are essential anchors when graphing equations. For \(y = |x| - 1\), strategic selection of x-values can make the task straightforward.

Choosing x-values like -2, -1, 0, 1, and 2 provides a clear overview:
  • At \(x = -2\), \(y = 2 - 1 = 1\)
  • At \(x = -1\), \(y = 1 - 1 = 0\)
  • At \(x = 0\), \(y = 0 - 1 = -1\)
  • At \(x = 1\), \(y = 1 - 1 = 0\)
  • At \(x = 2\), \(y = 2 - 1 = 1\)
Plotting these points will help form the distinctive 'V' shaped graph.
Symmetry
The symmetry in the graph of an absolute value function is a valuable property. For the function \(y = |x| - 1\), the graph is symmetric about the y-axis. This means that for every point \( (x, y) \), there is a corresponding point \( (-x, y) \).

This symmetry allows for a balanced and accurate graph. Recognizing this symmetry simplifies plotting, as you only need to calculate the y-values for x-values on one side and reflect them across the y-axis.
Transformation of Functions
Transformations modify the position or size of graph shapes from basic functions. In \(y = |x| - 1\), the graph results from a vertical shift of the basic absolute value graph.

This transformation moves the whole graph down by 1 unit, evidenced by subtracting 1 in the function.

Understanding transformations is crucial when predicting graph behavior. It allows you to quickly adjust for shifts, stretches, or compressions that a basic function's graph might undergo based on mathematical operations applied.

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Most popular questions from this chapter

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