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Chapter 0: Problem 17

Graph. (Unless directed otherwise, assume that "Graph" means "Graph by hand.") $$y=|x|$$

Short Answer

Expert verified
Graph is a V-shaped graph with vertex at (0,0).

Step by step solution

01

- Understand the Absolute Value Function

The function given is y = |x|, which represents the absolute value of x. The absolute value function produces non-negative results for any input x.
02

- Analyze the Behavior of y = |x|

For x >= 0, y = |x| simplifies to y = x. For x < 0, y = |x| simplifies to y = -x. This creates a V-shaped graph with a vertex at the origin (0,0).
03

- Plot Key Points

Identify and plot key points on the graph. For example, when x = -2, y = 2; when x = -1, y = 1; when x = 0, y = 0; when x = 1, y = 1; and when x = 2, y = 2.
04

- Draw the Graph

Draw the graph by connecting the plotted points. Use straight lines to connect points because the graph of y = |x| consists of two linear segments forming a V shape. The left segment is for x < 0 and the right segment for x >= 0.

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Key Concepts

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

graphing functions
Graphing functions is an essential skill in mathematics, particularly in algebra and calculus. When graphing a function, you're creating a visual representation of all possible solutions or points that satisfy the function's equation. This helps in understanding the function's behavior and identifying patterns. For the absolute value function, the graph visually shows how it produces only non-negative results. By plotting points and joining them with line segments, you can better grasp the shape and behavior of the function.
plotting points
Plotting points is the first step in graphing a function. A point consists of an x-coordinate and a y-coordinate, written in the form (x, y). To plot the point, you find the corresponding position on the x-axis and y-axis. For example, for the point (1, 1), you move one unit to the right along the x-axis and one unit up the y-axis. For the function y = |x|, some key points are (-2, 2), (-1, 1), (0, 0), (1, 1), and (2, 2). By plotting these points, you lay the groundwork for drawing the graph.
piecewise functions
The absolute value function is a piecewise function, meaning it is defined by different expressions based on the value of the input. For y = |x|:
  • If x >= 0, then y = x
  • If x < 0, then y = -x
This results in two different linear segments that together create the overall graph. Understanding piecewise functions involves recognizing that the function's rule changes based on the input's range, resulting in different behavior on different intervals of the domain.
linear segments
Linear segments are straight lines that make up certain parts of piecewise functions. For the absolute value function, each linear segment is defined based on the input value of x. The left segment for x < 0 follows the rule y = -x, creating a line that decreases as x moves leftward. The right segment for x >= 0 follows the rule y = x, creating a line that increases as x moves rightward. Combining these segments forms the familiar V-shape of the absolute value function. Transitioning from one segment to another with precision is key when drawing the graph of piecewise functions.

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Most popular questions from this chapter

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