# What does the coefficients A, B, C, and D to the graph y=D \pm A \cos(B(x \pm C))?

Dec 6, 2014

The general form of the cosine function can be written as

$y = A \cdot \cos \left(B x \pm C\right) \pm D$, where

$| A |$ - amplitude;
$B$ - cycles from $0$ to $2 \pi$ -> $p e r i o d = \frac{2 \pi}{B}$;
$C$ - horizontal shift (known as phase shift when $B$ = 1);
$D$ - vertical shift (displacement);

$A$ affects the graph's amplitude, or half the distance betwen the maximum and minimum values of the function. this means that increasing $A$ will vertically stretch the graph, while decreasing $A$ will vertically shrink the graph.

$B$ affects the function's period. SInce the cosine's period is $\frac{2 \pi}{B}$, a value of $0 < B < 1$ will cause the period to be greater than $2 \pi$, which will stretch the graph horizontally.

If $B$ is greater than $1$. the period will be less than $2 \pi$, so the graph will shrink horizontally. A good example of these is

http://www.regentsprep.org/regents/math/algtrig/att7/sinusoidal.htm

Vertical and horizontal shifts, $D$ and $C$, are pretty straightforward, these values only affecting the graph's vertical and horizontal positions, not its shape.

Here's a good example of vertical and horizontal shifts:

http://www.sparknotes.com/math/trigonometry/graphs/section3.rhtml