There are activation functions with some learnable parameters (search Google for learnable activation functions and you'll find some papers). But it's not particularly common to use them. Instead, the weights of the layer are learned, and that can (together with the activation function) represent very complicated functions.
I will note that it is quite common to use AutoML techniques, which try a variety of architectures (including different activation functions) to see which works best.
There are activation functions with some learnable parameters (search Google for learnable activation functions and you'll find some papers). But it's not particularly common to use them. Instead, the weights of the layer are learned, and that can (together with the activation function) represent very complicated functions.
I will note that it is quite common to use AutoML techniques, which try a variety of architectures (including different activation functions) to see which works best.