Again, we notice something peculiar about a horizontal transformation. Specifically, to describe f getting 4 times wider algebraically, we have f(x)\rightarrow f(\tfrac x4), which is less natural than if we have a function that becomes 4 times taller: f(x)\rightarrow4f(x).

Can we make sense of why f(\tfrac{x}q) increases the x-value of each point by a factor of q?

Let's interpret what the algebraic form of the transformation, g(x)=f(\tfrac x4), actually means:

We get g(1) by taking f(\tfrac14), g(4) by taking f(1), g(8) by taking f(2), and so on.
We see each output of g at a is the output of f at \tfrac a4.
Thus, the reason a horizontal stretch by a factor q is represented by \tfrac xq is because the value of g at every x is equivalent to the value that f ouputs q times closer to the x-axis.

Linear Transformations