An even function is defined by the property that, for all values of x in the function’s domain, the following holds true:
f(-x) = f(x)
This relationship implies that the function produces the same output for both x and its negative counterpart, -x. Because of this symmetry in its values, the graph of an even function is symmetric about the y-axis.
To visualize this, imagine plotting points on a Cartesian plane. If you have a point (a, b) on the graph of the even function, then you will also have the point (-a, b). This means that for every point on one side of the y-axis, there exists a corresponding point directly opposite it, maintaining a mirror-like reflection across the axis.
Common examples of even functions include:
f(x) = x^2, whose graph is a parabola opening upwards and is clearly symmetric about the y-axis.f(x) = cos(x), the cosine function, also exhibits y-axis symmetry as it repeats its values in a mirrored fashion over this axis.
In summary, the key takeaway is that the graph of an even function shows y-axis symmetry, which means it looks the same on either side of the vertical axis, providing a balanced and harmonious visual representation of the function.