Question

the graph of -2sin(2x=π/2)+3

Answer 1

Conditions

the graph of $-2\sin(2x=\pi/2)+3$

Please show steps

Solution

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

At first, let's consider the graph of a basic function:

$$y = \sin x$$

Then, let's construct a graph

$$y = \sin(2x)$$

Which is 2 times compressed compared to previous.

The next step is to construct:

$$y = \sin \left(2x + \frac{\pi}{2}\right)$$

The difference between this one and previous one is in shift to the left for $\frac{\pi}{2}$.

The last step is to make a final graph:

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

"-2" means that our graph become 2 times larger in $\gamma$ -axes (from [-1;1] to [-2;2]) and "-" means that the graph is inverted relative to $x$ -axes.

" +3" means that the graph is 3 points higher than previous:

Or, for $x$ from -10 to 10: