First off, this is not a homework question. I’m just confused. Please reference the diagram for electromagnetic radiation found in the Wikipedia article Electromagnetic radiation - Wikipedia. I’ve seen this same diagram in many different books. Notice that the electric and magnetic components are in phase with each other. The article even points this out. My thinking is that since a magnetic field is created by a moving electric field and vice versa, that these components should be out of phase with each other by a quarter of a wavelength. Thus, according to my thinking, when the electric field has reached its maximum (and therefore not moving) the magnetic field should then be zero. Then as the electric field reverses and starts going down, the magnetic field crosses the zero axis and goes from positive to negative. Thus, energy is continuously flowing back and forth between a magnetic field and an electric field with the total energy remaining constant as the wave travels through space. With the diagram that is usually shown, the total energy seems to appear and disappear as the wave travels through space. What am I missing? 
You are missing that electromagnetic waves are a single thing and not individual phonominia.
The electromagnetic force will be observed as either what we call an “magnetic field” or a “electrical field” based on the chosen reference frame.
If you know some math and can read the Maxwell Equations you will notice the second two are really related to geometry.
This 3Blue1Brown video produced a video that may help a bit with the intuitions here, but “energy is continuously flowing back and forth between a magnetic field and an electric field” is probably the easiest misconception for you to correct. There is no magnetic field or electric field, both those frame dependant observables arise from a single electromagnetic field
This forum doesn’t have the tools to even try and show why, but if you dig into this a bit more it is easy to see just how beautiful it is, especially how it relates to how the complex plane make the idea really elegant.
I think the idea is that, at least for linearly polarized electromagnetic waves, the electric and magnetic fields are in phase, just at right angles to each other.
That is a special case that make visualization and the math easy.
Euler’s identity or the orthogonality of the imaginary component from:
i: i×i+1×1=0
For the math side specifically, but this is a simplification but that orthogonal imaginary axis is critical to even have analytical solutions under our known models too.
In the case of EM it also has implications related to degrees of freedom and symmetry but as the SDMB lacks math display support it is pretty hard to explain.
Noether’s theorem’s implication with conserved properties with tensors of energy-momentum and angular momentum for the EM field in the presence of charges and currents would be something to look into if you wanted an example.
That orthogonal imaginary axis is critical for that but it does also relate to maintaining phase information in many operations that I can’t figure out how to put into words.
Just curious: are they ever out of phase?
Yes–for circularly polarized waves. The phase offset (whether +90 or -90 degrees) determines if it’s left-handed or right-handed. And really, this is still just a special case of elliptical polarization, where the phase offset can be anything.
In a vacuum, the electric and magnetic fields are always in phase (that is to say, passing through zero at the same points), regardless of polarization. This is, basically, because the temporal derivative of the electric field is proportional to the spatial derivative of the magnetic field, and vice-versa: In either case, it’s a derivative, and for a sine wave, a spatial derivative is basically the same as a time derivative.
I see that above I answered the wrong thing. It is the vertical and horizontal components of each of the electric and magnetic fields that can have an arbitrary phase offset, which then determines the polarization. But as Chronos says, the E and B fields relative to each other are always in phase (in a vacuum, at least).
Thanks Rat Avatar. That YouTube video was great. I’m also trying to understand Bell’s Theorem. Of the several explanations I’ve seen, I’m not convinced the people trying to explain it understood it themselves; their explanations sound goofy to me. Maybe it’s just beyond me math wise. Back in college (fifty years ago) I faked my way through the differential equations class. Gave up going further in math. Never needed anything more than basic algebra for any job I’ve had as a software engineer.
Or, to put it another way, the “point” at which the electric and magnetic fields are zero is a point traveling through both space and time. Pictures are misleading because they don’t move.