Understanding the Graph of the Absolute Value Function: Exploring the V-Shaped Symmetry and Slope Rates

graph of absolute value

The graph of the absolute value function, denoted as |x|, is a “V-shaped” graph that opens upwards or downwards

The graph of the absolute value function, denoted as |x|, is a “V-shaped” graph that opens upwards or downwards.

To understand the graph, we need to consider the two cases when x is positive and when x is negative.

1. When x ≥ 0 (x is positive):
In this case, the absolute value of x is equal to x itself. Therefore, the graph will simply be a straight line passing through the origin (0,0) and increasing as x increases. It forms a “V” shape that opens upwards.

2. When x < 0 (x is negative): In this case, the absolute value of x is equal to -x. Thus, the graph will also be a straight line, but it will be reflected across the x-axis. It will start from the origin and decrease as x becomes more negative. This forms a "V" shape that opens downwards. Together, both cases result in a symmetric "V" shape, with the vertex at the origin. It is important to note that the point (0,0) is always included on the graph since the absolute value of zero is zero. To summarize: - For x ≥ 0, the graph is a straight line that increases as x increases. - For x < 0, the graph is a straight line that decreases as x becomes more negative. - The graph is symmetric with respect to the y-axis and its vertex is at the origin (0,0). Here is a visual representation of the graph of |x|: ``` | | /\ ______|____/__\_____ | / \ | ``` Note that the edges of the "V" shape have a slope of 1, indicating the rate of change for positive and negative values of x.

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