Trigonometric graphs

Functions

$\displaystyle f(x) = a \sin (bx + c)$    and $\displaystyle \quad f(x) = a \cos (bx + c)$ (1-2)

have (i) the amplitude $ \vert a\vert$, (ii) the period  $ \dfrac{2\pi}{\vert b\vert}$, and (iii) the phase shift  $ -\dfrac{c}{b}$. Notice that (1-2) can be rewritten in the respective form of (1-1) with $ \omega = \vert b\vert$ and $ \phi = \dfrac{c}{b}$:

$\displaystyle f(x) = a \sin (bx + c) = \begin{cases}a \sin \vert b\vert\left(x ...
...in \vert b\vert\left(x + \dfrac{c}{b}\right) & \mbox{ if $b < 0$, } \end{cases}$ (1-3)

$\displaystyle f(x) = a \cos (bx + c) = a \cos \vert b\vert\left(x + \dfrac{c}{b}\right).$ (1-4)

Cycle. The interval  $ [ -c/b,  -c/b+2\pi/\vert b\vert ]$ represents one cycle. The graph of one cycle is given by the functions (1-3)-(1-4) with phase shift $ c = 0$.



Department of Mathematics
Last modified: 2005-09-29