Question

How to understand `y=f(2x)` ? Why is this really a compression? If `f(x) = 2x+3` can someone show a table for this? In my text book it says we divide the `x` value by `2`? I don't see why? Wouldn't we times it by `2`? Thanks.

Answer 1

Initially we may naively think that the graph of `y=f(2x)` is a stretch of `y=f(x)`. However, we can show that in fact it is a compression, if we construct a table for a generic function for some (arbitrary) integer values of the domain:

`{: (x,0,1,2,3,4,5,6), (f(x),color(red)(f(0)),f(1),color(green)(f(2)),f(3),color(blue)(f(4)),f(5),color(purple)(f(6))), (,,,,,,,), (f(2x),color(red)(f(0)),color(green)(f(2)),color(blue)(f(4)),color(purple)(f(6)),f(8),f(10),f(12)) :}`

And, we can see that over the same same scale the graph of `y=f(x)` would be a stretch of the graph of `y=f(2x)`, or conversely the graph of `y=f(2x)` would be a compression of the graph of `y=f(x)`.

We can readily verify this behaviour with a graph, first of the given functions`y=f(x)` then `y=f(2x)` where `f(x)=2x+3`

graph{2x+3 [-2, 7, -2, 30]}
graph{2(2x)+3 [-2, 7, -2, 30]}

And for this function, we can compute some values:

`{: (x,0,1,2,3,4,5,6), (f(x),color(red)(3),5,color(green)(7),9,color(blue)(11),13,color(purple)(15)), (,,,,,,,), (f(2x),color(red)(3),color(green)(7),color(blue)(11),color(purple)(15),19,23,27) :}`

And next with the periodic function `y=f(x)` then `y=f(2x)` where `f(x)=sinx` where we can clearly see the period of the later function is twice that of the former due to the compression:

graph{sinx [-10, 10, -2, 2]}
graph{sin(2x) [-10, 10, -2, 2]}