Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = -2|x|

Verified step by step guidance

1. Step 1: Understand the equation y = -2|x|. This equation involves the absolute value function |x|, which outputs the non-negative value of x. The negative sign in front of the 2 indicates that the graph will be reflected across the x-axis, and the coefficient -2 indicates a vertical stretch by a factor of 2.
2. Step 2: Create a table of values for the given x-values (-3, -2, -1, 0, 1, 2, 3). For each x-value, calculate |x| (the absolute value of x), then multiply it by -2 to find the corresponding y-value. For example, when x = -3, |x| = 3, and y = -2(3) = -6.
3. Step 3: Fill in the table with the calculated values. For example: x = -3, y = -6; x = -2, y = -4; x = -1, y = -2; x = 0, y = 0; x = 1, y = -2; x = 2, y = -4; x = 3, y = -6.
4. Step 4: Plot the points from the table on a coordinate plane. Each point corresponds to an (x, y) pair, such as (-3, -6), (-2, -4), (-1, -2), (0, 0), (1, -2), (2, -4), and (3, -6).
5. Step 5: Connect the points with straight lines to form the graph. The graph will have a V-shape, opening downward due to the negative coefficient, and it will be symmetric about the y-axis because the absolute value function is symmetric.

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, measures the distance of a number x from zero on the number line, always yielding a non-negative result. For example, |3| = 3 and |-3| = 3. This function is crucial in understanding how the graph behaves, particularly in reflecting values across the x-axis.

Linear Transformations

Linear transformations involve modifying the basic shape of a graph through scaling, translating, or reflecting. In the equation y = -2|x|, the negative sign indicates a reflection over the x-axis, while the coefficient -2 scales the graph vertically, making it steeper. Understanding these transformations helps in accurately sketching the graph.

Graphing Techniques

Graphing techniques involve plotting points and understanding the shape of the graph based on the equation. For y = -2|x|, we calculate y-values for given x-values, which helps in visualizing the V-shaped graph that opens downward. Mastery of these techniques is essential for accurately representing equations on a coordinate plane.