Equivalent horizontal and vertical transformations of functions

Question

I have been investigating equivalent transformations of functions, particularly those relating horizontal scaling to vertical scaling. For example, $f(x) = a\sqrt {bx} = a\sqrt{b}\sqrt{x}$, so horizontally scaling the function $g(x) = a \sqrt{x}$ by a factor of $\frac{1}{|b|}$ is the same as vertically scaling the function by a factor of $\sqrt{b}$. We can clearly derive similar equivalent transformations for $y=a(bx)^2$, $y = \frac{a}{bx}$, etc.

However, I have not been able to apply similar reasoning for functions like $y=a\sin{bx}$ or $y = a2^{bx}$. Of course, it is sometimes possible using identities ($y = \sin{2x} = 2\sin{x}\cos{x}$ and $y = \log{bx} = \log{x} + \log{b}$), but I am interested in a more general conclusion, particularly one relating horizontal and vertical scaling. Is there such a result, and does it reveal anything fundamental or interesting about functions? Or, is there a reason that such a result would not exist or would be trivial? Perhaps there is another way to view this problem?

Answer 1

It's impossible for $y = a \sin bx$ because vertical scaling controls the amplitude of the wave, while horizontal scaling controls the period of the wave: they are independent of each other.

In general, you can't substitute horizontal scaling with vertical scaling.

Answer 2

The main transformations are

• $y = f(x) + C \implies C > 0$ moves it up, $C < 0$ moves it down

• $y = f(x + C) \implies C > 0$ moves it left $C < 0$ moves it right

• $y = Cf(x) \implies C > 1$ stretches it in y-direction, $0 <C<1$ compresses it

• $y = f(Cx) \implies C > 1$ compresses it in the x-direction,$0 < C < 1$ stretches it

• $y = −f(x)$ reflects it about x-axis

• $y = f(−x)$ reflects it about y-axis

• $y=|f(x)|$ reflects negative values of f(x) about x-axis