General Sinusoidal Graphs

Key Questions

• Suppose the graph is of the form y = A sin (bx + c) + d
Then the amplitude is A and represents the maximum value of the range on the y axis. The period can be given by 2pi/b and represents the number of radians on the x axis for a complete cycle of the curve. The c value represents the phase angle shift and indicates how many radians the graph is shifted left or right from the origin. The d represents a vertical transformation and indicates how many units the graph range is shifted away from the amplitude at the origin.

• Example: Describe the transformations to get $g \left(x\right) = 2 \sin \left(3 \left(x + \frac{\pi}{4}\right)\right) + 2$ from $f \left(x\right) = \sin x$

Here are what each of the parameters in the equation $y = a \sin \left(b \left(x - c\right)\right) + d$:

$a \to$ vertical stretch
$\frac{1}{b} \to$horizontal stretch
$c \to$ phase shift
$d \to$vertical transformation

So in the given equation, we have a vertical stretch by a factor of $2$, a horizontal stretch by a factor of $\frac{1}{3}$, a transformation $\frac{\pi}{4}$ units left and a transformation $2$ units up.

Hopefully this helps!

• 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

• "Sinusoidal" means a movement that goes in one direction, then gradually slows down, stops and changes into an opposite direction, then again slows down, stops and changes into an original direction. This cycle repeats itself.

Presented on the time line (X-axis) with directions of movement along the Y-axis, this behavior resembles the graph of a trigonometric function $y = \sin \left(x\right)$. That's why the name of this behavior.

The movement described above is not necessarily a mechanical movement. It can be a change in the temperature, pressure, even the person's mood going from excitement to depression and back.