What does it mean sum and multiply by 0.5?

Question

Sorry for the simple question but it's not clear to me.

Suppose I have this curve:

y = (0.5 * cos(period * (x - phase)) + 0.5)

what does it mean sum and multiply by 0.5?

I mean, 0.5 is in degree (not radians) and what is the difference between y = (0.5 * cos(period * (x - phase)) + 0.5) and y = cos(period * (x - phase))?

Answer 1

It's a simple trick to turn a function with a range from $-1$ to $1$ into a function with range from $0$ to $1$ (remember that sine and cosine functions go from $-1$ to $1$):

$f(x) \in [-1, 1] \iff \frac{f(x)}{2} \in [-\frac{1}{2}, \frac{1}{2}] \iff \frac{f(x)}{2} + \frac{1}{2} \in [0,1]$

Answer 2

Summing is to move the graph up. In your case it's moving up the graph by 0.5. Multiplying is to change the amplitude of the graph. In your case it's halving the amplitude.

You can analyse a cosine function inside out, by using the parent function $\cos(x)$. For example, $3\cos(x-\frac{\pi}{2})+2$ is the graph of $\cos(x)$:

1. shifted by $\frac{\pi}{2}$ to the right,
2. amplified by 3 times, and
3. shifted upwards by 2 units.

Answer 3

To explain multiply by 0.5 I suggest to have a look at the Intercept theorem.

That's for the multiplication. To sum is simply a translation.