What does the axis of a curve for a cosine function mean in a word problem? For example, I have the equation $$y = 8\cos\left[12(x-30)\right] + 10,$$ which represents the vertical displacement of a windmill. The axis of the curve is $y = 10$, but in real life terms, what does this mean?
The given height of the windmill is 18m, and it takes 30s to go from 18m at the start, down to 2m off the ground, and the 18m again.
On
To go beyond just the the vertical consideration of the wave, $A, B, C, $and$ D$ of the general formula $y = A \cos(Bx + C) + D$ are as follows:
Your formula is $y = 8 \cos(12x - 360)+10$
A = amplitude........in your case 8 which is the wave height above and below the horizontal center axis.
B = period = $\frac{360}{B}$..........in your case $\frac{360}{12} = 30$. When $B=1$, period $= 360$
C = horizontal location = $\frac{B}{C}$........in your case $-360/12$ is shifting the peak where $y = 18$ from $0$ to the right by $30$. Negative to the right, positive to the left.
D = vertical location........in your case $10$ shifts the horizontal center axis from $0$ to $10$
To answer this, I need to make a bunch of assumptions about your problem, like "the vertical displacement of the windmill" really means the height of the endpoint of one of the windmill blades".
Further assuming that the base of the windmill is at height $0$, and that $y$ is measured in, say, meters, and that $x$, rather than representing horizontal position, represents time, then the "$10$" indicates that the average vertical displacement --- the height of the hub of the windmill --- is 10 meters.