# How do I graph sinusoidal functions?

Aug 2, 2015

See explanation.

#### Explanation:

We know what the basic graph of $y = \sin x$ look like.

graph{y = sinx [-12.11, 16.36, -6.92, 7.31]}

The graph goes up to a maximum of $1$ and down to a minimum or $- 1$. It is 'centered' on the $x$ axis which is the line $y = 0$.

Let's look at: $f \left(x\right) = D + A \sin \left(B x + C\right)$

The number $D$ will shift the entire graph up (if $D > 0$) or down ($D < 0$) by the amount $\left\mid D \right\mid$
If $D \ne 0$, the center line will move to the horizontal line: $y = D$

Let's leave that vertical shift out and just look at:

$y = A \sin \left(B x + C\right)$

If $B < 0$, use the fact that $\sin \left(- u\right) = - \sin \left(u\right)$ to rewrite with a positve coefficient of $x$.

If $A$ is negative, we will reflect the graph across the $x$ axis.

Multiplying the sine by $A$ makes the new graph have a maximum of $\left\mid A \right\mid$ and a minimum of $- \left\mid A \right\mid$.
The number $\left\mid A \right\mid$ is called the Amplitude of the graph.

The period of the graph is the length of one complete cycle through the graph. For the basic graph, the period is $\pi$.
We often think of and describe this by saying, "The first period of the graph starts when we take the sine of $0$ and ends when we take the sine of $2 \pi$. That means the angle has gone once around the circle."

Returning to $y = A \sin \left(B x + C\right)$

We will take the sine of $0$ when $B x + C = 0$
and we will take the sine of $2 \pi$ when $B x + C = 2 \pi$

So we will start the first period when $x = - \frac{C}{B}$ (solve the first equation.)
The number $- \frac{C}{B}$ is called the Phase (or Horizontal) Shift.

And we will finish the first period when $x = \frac{2 \pi - C}{B}$ (Don't memorize that, we have another way of finding it.)

The "length of the period" is the end minus the start, or:

$\left(\frac{2 \pi - C}{B}\right) - \left(- \frac{C}{B}\right)$ And that simplifies to $\frac{2 \pi}{B}$

So that is the why, and here is the summary:

$y = A \sin \left(B x + C\right)$ has

Amplitude: $\text{ "absA" }$ (That's how far above and below the center line we need to go.)

Period: $\text{ "(2pi)/B" }$

Phase Shift: $\text{ }$ Is the solution to $B x + C = 0$