# How do you sketch the graph of f(x) = 3sin^2(2x + Pi/4)?

Jul 28, 2018

See the graph of this sine wave and explanation.

#### Explanation:

$0 \le y = 3 {\sin}^{2} \left(2 x + \frac{\pi}{4}\right) \in 3 \left[0 , 1\right] = \left[0 , 3\right]$

$y = 3 {\sin}^{2} \left(2 x + \frac{\pi}{4}\right) = \frac{3}{2} \left(1 - \cos \left(2 \left(2 x + \frac{\pi}{4}\right)\right)\right)$

$= \frac{3}{2} - \frac{3}{2} \cos \left(4 x + \frac{\pi}{2}\right) = \frac{3}{2} + \frac{3}{2} \sin 4 x$.

So, this a sine wave.

The period $= \frac{2 \pi}{4} = \frac{\pi}{2}$.

The amplitude $= \frac{3}{2}$.

The axis is # y = 1.5

Now see the wave, withe these aspects duly depicted.
graph{(y-3/2 - 3/2 sin (4x) )(y-3+0x)(y-1.5+0x)(x-1/2pi+0.0001y)(x+1/2pi+0.0001y) = 0[-3.4 3.4 -0.2 3.2]}