Explain what all the parts of the vertex form of the quadratic equation f(x)=a(x-h)^2+k

Can someone answer it, and plot it, for 20 points and brainliest answer? p.s. they're the same ! : )

Asmall (but heavy) particle placed in a glass of water will follow a zigzag motion because the particle will bounce off of the water molecules it meets. this is called brownian motion. a physicist simulates this on a computer, by varying the distance a particle can travel (called the mean free length), on average, before it collides with a water molecule and assigning the change in motion to be one of 8 directions, each with a similar probability. by running the simulated particle (with the same mean free length) many times she determines that it should take 15 seconds, on average, for the particle to fall to the bottom, with a standard deviation of 1.5 seconds. next she lets a real particle fall through a glass of water and finds that it took 18 seconds. what does she conclude, and why?

If cos(2x) = tan^2(y), show that cos(2y) = tan^2(x).