In-Depth on Function Transformations

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If all functions can be written in the form y = af(x-h)+k, how does this account for horizontal stretching and compression? Why is the default form for all functions not y = af(b(x-h))+k. I know of two functions that require the second notation which are sine and cosine. Are there any other functions like this? The reason I ask this question is because most times when finding an equation based on points, you do not have enough points to also account for horizontal compression and stretching. Are sine and cosine simply exceptions to the rule?