There are certain properties that turn out to be really helpful and exploitable - especially in physics and calculus courses. We discuss one of these here, the so-called “even” functions.

Lecture Video

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There are two main ways to try and understand the “even property”, and it comes down to the fact that it is essentially a type of symmetry. We begin with the geometric view - explicitly discussing the symmetry, and then continue on to the analytic view to nail down the corresponding algebraic property, which will ultimately give us the formal definition.

Geometric Perspective

Consider the following graph:

Here we have the classic parabola, but for a moment ignore the math that you know and just look at the picture of the curve. It has a pleasing smoothness and it is perfectly centered horizontally - indeed we could imagine carefully folding along the $y$-axis and watching as the lines match up with a satisfying precision, overlapping perfectly.

This kind of symmetry is called axial symmetry. The $y$-axis is our line of symmetry and if we rotate the picture around it 180 degrees, we get the same exact image back. This is a general geometric property, but for the symmetry to be a result of the “even property”, we need that the line of symmetry must be the $y$-axis. This is the case in our previous graph, but consider the following graph:

Although this is essentially the same curve, just shifted to the right slightly, we can’t say that this graph has the “even property” because the line of symmetry has moved. To be clear, we could say that this function is “symmetric about the line $A$” - but this is not the same as saying that the “graph is even” or “has the even property”.

Analytic Perspective

So now we can visualize the even property as axial symmetry about the $y$-axis. But how does this translate into an algebraic property that we can use to represent this type of symmetry? Let’s return to our original example, but this time we draw only the right half.

Now, on the one hand, we are missing half the graph, but on the other hand, we know what the missing half should look like right? Thanks to the symmetry, we can use the right half of the graph to produce the missing half - by simply copying what we see on the right, over to the left:

But, how does this get us closer to the algebraic property we want? Well let’s consider what just actually happened. In essence, the points to the right of the $y$-axis are the same as the ones we are getting to the left of the $y$-axis - the only difference is that we are going left instead of right. In other words, if there is a point $(x,y)$ on the right side somewhere (i.e. a point “$x$” distance to the right and “$y$” distance up), then the point $(-x,y)$ (i.e. “$x$” distance to the left and “$y$” distance up) is also on the graph.

As a concrete version of this, consider our mirrored graph again, but let’s pick a point to the right of the $y$-axis somewhere - in particular we’ll choose $(2,1)$ and label it.

Because of the symmetry, we know that a mirror point has to exist on the left side of the $y$-axis - which means that the $y$ value has to stay the same, and the $x$ value is the same size, but in the other direction from the $y$-axis.

This is the key observation - that we can flip the $x$ value (meaning the same size, but the opposite direction) and the $y$ value remains the same. Importantly it doesn’t actually matter which of the points we did this with either - we could have started with the point $(-2,1)$ and flipped the $x$ value to get $(- (-2),1) = (2,1)$ which is still a point on the graph. The key isn’t whether or not the $x$ value itself is positive or negative - the key is that both the positive and negative version have the same $y$ value!

This leads us to the analytic property for this kind of symmetry - that if $(x,y)$ is a point on the graph, then the point $(-x,y)$ must also be a point on the graph. It turns out, for reasons that are beyond the scope of this course, we call functions with this type of symmetry “even” functions, and this property is known as the “even property”.

The definition $f(x)=f(-x)$ is usually the part students remember (and it is indeed the useful algebraic result), but the geometric understanding is usually easier to understand. The real takeaway here is the idea that “if you go the same distance left or right from the $y$-axis, you get the same $y$ value.”