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To graph an absolute value function like \[ y = 2|x| \], you'll first want to plot points. Start with choosing a set of x-values, like \(-3, -2, -1, 0, 1, 2, 3\). Calculate the corresponding y-values by plugging each x-value into the function. Here’s what you get:

• For x = -3, y = 2|-3| = 6
• For x = -2, y = 2|-2| = 4
• For x = -1, y = 2|-1| = 2
• For x = 0, y = 0
• For x = 1, y = 2
• For x = 2, y = 4
• For x = 3, y = 6

After calculating the values, plot each pair on a coordinate plane. You'll have points like (-3, 6), (0, 0), and (3, 6). Connecting these points will give you the shape of the absolute value function. Remember that for any absolute value function, the graph will form a 'V' shape.

Determining the domain and range of the function \[ y = 2|x| \] is an essential part of graphing. The domain refers to all the possible x-values that you can input into the function. For absolute value functions, you can input any real number, so the domain is \((-\infty, \infty)\), which means all real numbers.
The range refers to all the possible y-values that can result from the function. Since \[ y = 2|x| \] multiplies the absolute value of x by 2, the smallest value y can take is 0, which occurs when x is 0. As x moves away from 0 in both the positive and negative directions, y increases. Therefore, the range is \([0, \infty)\), which includes zero and all positive real numbers.

An absolute value function graphs a special kind of equation that involves the absolute value operation, denoted as |x|. The absolute value of a number is its distance from zero on the number line, without considering direction. For example, |3| = 3 and |-3| = 3.
In our given absolute value function \[ y = 2|x| \], we see a V-shaped graph. The vertex or the lowest point of this graph is at (0,0). The graph opens upwards because the absolute value function returns non-negative values, and the factor of 2 outside the absolute value means the graph is vertically stretched by a factor of 2 compared to \ |x| \.
Understanding the role of the absolute value and any constant multiplication is crucial. Here are some key points:

• The absolute value creates a V-shape because it reflects all negative x-values into positives.
• The number outside the absolute value, in this case, 2, affects the stretching. A larger number makes the V-shape narrower, while a smaller number would make it wider.