Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problema 22a

Capitolo 2, Problema 22a

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

Guida verificata passo dopo passo

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.

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.

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.

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).

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.

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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.

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Function Composition

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.

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Intro to Transformations

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.

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Graphs and Coordinates - Example

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