Understanding the Absolute Value Function: Definition, Graphing, and Symmetry

f(x)=|x|

The function f(x) = |x| is known as the absolute value function

The function f(x) = |x| is known as the absolute value function.

The absolute value function represents the distance of a number from zero on the number line. It essentially “absolutizes” any input value, making it positive.

The function f(x) = |x| can be defined as follows:

When x is greater than or equal to zero (x ≥ 0), the absolute value of x is equal to x itself. So, f(x) = x.

When x is less than zero (x < 0), the absolute value of x is equal to the opposite of x, or -x. So, f(x) = -x. To graph the function f(x) = |x|, you can follow these steps: 1. Draw a coordinate plane with the x-axis and y-axis. 2. Take a few sample values of x to calculate f(x) and plot the corresponding points on the graph. For example: - If x = 0, then f(x) = |0| = 0. So, plot the point (0, 0). - If x = 1, then f(x) = |1| = 1. So, plot the point (1, 1). - If x = -1, then f(x) = |-1| = 1. So, plot the point (-1, 1). - Repeat this process for more values of x, keeping in mind that the absolute value of a negative number is the opposite of that number. 3. Once you have plotted enough points, connect them using a smooth curve. The resulting graph will be a V-shape, with the vertex at (0, 0). The graph is symmetrical about the y-axis, meaning that for every positive x-value, there is a corresponding negative x-value with the same y-value. In summary, the absolute value function f(x) = |x| represents the distance of a number from zero and results in a V-shaped graph that is symmetrical about the y-axis.

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