How can I get a cosine-looking graph with the following requirements?

[dreamy trout]

I want the graph to resemble y = cos(x) + 1 at first glance,
however as |x| increases the amplitudes of the graph gradually decrease.
I want the minima of the graph to remain where they are (along the line y = 0)

What function would look like this

[abstract palm]

you could multiply the function you already have by any function which decreases slowly as |x| increases

many options for that decreasing function. 0.95^|x| for example

[pearl bridge]

For large $V$, $e^{-\frac{x^{2}}{V}}\left(\cos\left(x\right)+1\right)$ should work

[sonic ice]

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