I'm trying to make a model of the rise and fall of sea levels. According to this explanation and image in the textbook, the product of two sinusoidal functions should look something like this:
(Note that the graph's equation is written next to point #6.)
However when I used the same method of finding a product function for the sea level data, I got this
(Note that the blue and green are the two functions and the yellow is the product function.)
In other words, the graph in the textbook has an equal amplitude throughout the entire x-axis, however the function that I came up with changes amplitude in the second half of the x-axis. Why is this?
According to a comment, you are multiplying the function $$f(x)=2\sin(22x)-0.4$$ by the function $g(x) = \cos(5x).$
The function $f(x)$ has a much longer period than $g(x)$, but it also has unequal amplitude above and below the axis. That is, the highest possible value of $f(x)$ is $$2-0.4 = 1.6$$ and the lowest possible value is $$-2-0.4 = -2.4$$
While $f(x)$ varies very slowly between $1.6$ and $-2.4$, $g(x)$ varies rapidly between $-1$ and $1$. In the left half of the diagram, there are some cycles of $g(x)$ during which $f(x)$ is about $1.5$ (and never greater than $1.6$), so the product $f(x)g(x)$ varies between about $1.6 \times -1 = -1.6$ and $1.6 \times 1 = 1.6$.
But in the right half of the diagram, there are some cycles of $g(x)$ during which $f(x)$ is about $-2.4$, so the product $f(x)g(x)$ varies between about $-2.4 \times 1 = -2.4$ and $-2.4 \times -1 = 2.4$.
A wave that varies from $-2.4$ to $2.4$ has a greater amplitude than a wave that varies from $-1.6$ to $1.6$, so the perception that the amplitude is greater in the second half than in the first half is accurate.
In the figure in the book, and in the exercise beneath it (plotting $y = 5 \sin \theta \cos (11\theta)$), the two functions to be multiplied are both unbiased (symmetric above and below the horizontal axis), which is why the product appears to have the same maximum amplitude in both halves of the figure.
Note: The formulas above do not actually match the graph, but these are the functions OP wishes this answer to deal with, and the analysis of these functions is still applicable to the original graph, albeit with somewhat different constants.