The function y equals absolute value of x is symmetric because it's an even function. A function is symmetric about the y-axis when f of negative x equals f of x for all x values. Let's visualize this property by examining the graph of y equals absolute value of x.
To verify that y equals absolute value of x is an even function, we need to check if f of negative x equals f of x. Let's test this with x equals 2. We have f of negative 2 equals absolute value of negative 2, which equals 2. And f of 2 equals absolute value of 2, which also equals 2. Since both points have the same y-coordinate, this confirms the even function property.
Now let's prove this property generally for any value of x. For the function f of x equals absolute value of x, we need to show that f of negative x equals f of x. We have f of negative x equals absolute value of negative x. Since the absolute value of negative x equals the absolute value of x for any real number, we get f of negative x equals absolute value of x, which equals f of x. This animation shows how any pair of opposite x-values always produces the same y-value, confirming the symmetry.
The symmetry of y equals absolute value of x can be visualized as a perfect reflection across the y-axis. The left side of the graph is a mirror image of the right side. Each point on the left branch at coordinates negative a, absolute value of a, has a corresponding point on the right branch at coordinates a, absolute value of a, both at exactly the same height. This mirror symmetry across the y-axis is the geometric representation of the even function property, confirming that y equals absolute value of x is indeed a symmetric function.
In conclusion, y equals absolute value of x is a symmetric function because it satisfies all the requirements of an even function. We proved that f of negative x equals f of x, demonstrated that the absolute value of negative x equals the absolute value of x, and showed that the graph exhibits perfect reflection symmetry across the y-axis. Therefore, y equals absolute value of x is definitively an even function with complete symmetry about the y-axis.