Introducing all possible parameters into the sine function we get: $$f(x) = A\sin( kx - \phi),$$ where $A$, $k$ and $\phi$ are the parameters.

• $A$ is the amplitude, which tells you the distance the function will go above and below the $x$ axis as it oscillates.
• $k$ is the wave number and decides how many times the graph goes up and down within one period of $2\pi$. For the “bare” sine, $k=1$ and the function makes one cycle as $x$ goes from $0$ to $2\pi$. If $k=2$ the function will go up and down twice.
• $\phi$ is a phase shift, analogous to the horizontal shift $h$ which we have seen. This is a number which dictates where the oscillation starts. The default sine function has zero phase shift ($\phi=0$), so it starts from zero with an increasing slope.

Instead of counting how many times the function goes up and down, we can instead talk about the wavelength of the function: $$\lambda \equiv \text{ wavelength} = \{ \text{ the distance form one peak to the next } \}.$$ The “bare” sine has wavelength $2\pi$, but when we introduce some wave number multiplier $k$, the wavelength becomes: $$\lambda = \frac{2\pi}{k}.$$