# How do you find the amplitude, period, and shift for f(x) = −4 sin(2x + π) − 5?

Feb 12, 2016

First, let us make this equation in a different format
$f \left(x\right) = A \sin \left(B \left(x + C\right)\right) + D$

Only the inside of the sin must change.
We get this to be $2 \left(x + \frac{\pi}{2}\right)$

Now we have $f \left(x\right) = - 4 \sin \left(2 \left(x + \frac{\pi}{2}\right)\right) - 5$

The amplitude is always the absolute value of the coefficient of sin

This means that the amplitude is $4$

The period can be denoted as $\frac{2 \pi}{b}$ where b is the coefficient inside the sin

So we have $2 \frac{\pi}{2}$ ===> $\pi$

Now the vertical shift is easy: it is just the D value, which is a vertical shift down by 5 units

The horizontal shift is somewhat tricky. It is related to the C value
Similar to radical functions, this horizontal shift is the opposite of the sign.

For example: $\left(x - \pi\right)$ would be a horizontal shift to the RIGHT by $\pi$

and vice versa

Can you find the horizontal shift after knowing this?

It is a shift to the LEFT by $\frac{\pi}{2}$