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Chapter 1: Problem 21

Graph each equation . Let \(x=-3,-2,-1,0\) \(1,2,\) and 3. $$y=2|x|$$

Short Answer

Expert verified
The graph of \(y = 2 |x|\) is a V-shaped graph which passes through the following points: (-3, 6), (-2, 4), (-1, 2), (0, 0), (1, 2), (2, 4) and (3, 6).

Step by step solution

01

Identify Absolute Value

First, identify the absolute value in the equation \(y = 2|x|\). The absolute value always gives a positive output or zero. Therefore, the y-values, obtained by substituting the x-values in \(y = 2|x|\), will either be positive or zero.
02

Substitute the X-values

Substitute the given x-values in the equation \(y = 2 |x|\) to get the corresponding y-values. For x=-3, -2, and -1, the absolute value will transform these into 3, 2, and 1, giving y-values as 6, 4, and 2, respectively. For x=0, the y value is 0. For x=1, 2 and 3, the y-values will again be 2, 4 and 6, respectively.
03

Create Points

Now, create points for each x-y pair (x, y). These points are (-3, 6), (-2, 4), (-1, 2), (0, 0), (1, 2), (2, 4) and (3, 6).
04

Plot the Points

In this step, plot all of the points from the previous step on a graph. Connect them while keeping in mind that the graph of the absolute function has a 'V' shape, due to the nature of the absolute function.
05

Add Labels

Finally, make sure to label the x and y-axis and the points on them. Also, include the formula \(y = 2|x|\) with the graph.

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Key Concepts

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

Absolute Value
In mathematics, an absolute value represents the distance of a number from zero on the number line. It is always a non-negative number. For example, the absolute value of
  • \(-3\) is \(3\)
  • \(0\) is \(0\)
  • \(5\) is \(5\)
This feature of being non-negative makes absolute values particularly useful in various mathematical equations where you want to eliminate negative numbers.

In the equation given, \(y = 2|x|\), the absolute value symbol \( |x| \) indicates that for any input \(x\), the output will never be negative. This is a crucial aspect when determining the graph of the function, as all calculated \(y\)-values are zero or positive, creating a V-shaped result.
Graphing
To graph absolute value functions like \(y = 2|x|\), it's essential to understand how the graph's shape forms. Most absolute value graphs have a characteristic 'V' shape. This shape occurs because absolute values always yield non-negative results.

The 'V' shape emerges as points plot symmetrically around the vertical axis which is in the middle of the graph. By evaluating points to the left and right of zero, you see identical, yet opposite, results that reflect across the y-axis. This symmetric nature is a hallmark of absolute value graphs. Knowing this, drawing the graph becomes straightforward once you have plotted a series of calculated points.
Plotting Points
Once the x- and y-values are known, it's time to plot these points on a graph. In the exercise, you substitute specific x-values into the function \(y = 2 |x|\) to discover the corresponding y-values:

  • For \(x = -3, y = 6\)
  • For \(x = -2, y = 4\)
  • For \(x = -1, y = 2\)
  • For \(x = 0, y = 0\)
  • For \(x = 1, y = 2\)
  • For \(x = 2, y = 4\)
  • For \(x = 3, y = 6\)
These ordered pairs such as \((-3, 6)\), \((0, 0)\), and \((3, 6)\) are plotted on a Cartesian coordinate system grid.

Once you plot all the points, connect them with straight lines or curves, keeping in mind it forms a 'V' shape. It's equally important to label axes and points to ensure a clear understanding of the graph. Remember, labeling aids in interpreting and verifying that your graph represents \(y = 2|x|\) correctly.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=x^{3}\) and \(g(x)=-(x-3)^{3}-4,\) then the graph of \(g\) can be obtained from the graph of \(f\) by moving \(f\) three units to the right, reflecting about the \(x\) -axis, and then moving the resulting graph down four units.

If \(f(x)=x^{2}+3 x+2,\) find \(\frac{f(x+h)-f(x)}{h}, h \neq 0,\) and simplify. (Section \(1.3,\) Example 8 )

How is the standard form of a circle's equation obtained from its general form?

In Exercises \(67-70,\) graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{aligned} (x-2)^{2}+(y+3)^{2} &=4 \\ y &=x-3 \end{aligned}$$

What is the graph of a function?
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