Even functions have special symmetry that makes them predictable, especially when we look at the y-axis.

An even function has a simple rule:

[ f(-x) = f(x) ]

This rule is true for all values of ( x ) in the function's domain. Here’s what that means:

1. Graphical Symmetry:

   • If you have an even function and you pick a negative number, the output will be the same as for its positive counterpart.
   • This means that the graph will look the same on both sides of the y-axis.
   • For example, if you have a point like ( (a, f(a)) ) on the graph, you will also find the point ( (-a, f(a)) ) on the graph.

2. Examples:

   • Take the function ( f(x) = x^2 ).
     • For example:
     • ( f(2) = 4 ) and ( f(-2) = 4 ).
     • This shows it is symmetrical around the y-axis.
   • Another example is ( f(x) = \cos(x) ).
     • Here, ( \cos(-x) = \cos(x) ), so its graph also reflects across the y-axis.

3. Characteristics of Even Functions:

   • When it comes to polynomials, even functions usually have even degrees (like 0, 2, 4, etc.).
   • They can also involve even roots, like ( \sqrt{x} ) when we only consider non-negative values. But remember, they still need to follow the rule ( f(-x) = f(x) ).

4. Applications:

   • You’ll often find even functions in physics and engineering, especially when dealing with waves or symmetry.
   • The symmetry about the y-axis makes it easier to analyze their behavior. This helps when calculating areas and integrals, especially when things are symmetric around the center.

In short, the symmetry of even functions makes it simpler to draw and understand graphs in math, especially at the AS-Level.