# How do you graph f(x) = 4 sin(x - pi/2 ) + 1?

Mar 13, 2016

You have to identify a few important elements of the graph first.

#### Explanation:

Amplitude: in a function of the for $y = a \sin b \left(x + c\right) + d$, the amplitude is at $| a |$. So, the amplitude is at 4. The amplitude is the distance between the maximum and minimum points and the horizontal line of rotation. This line of rotation would be 0 if d = 0, but since there is a vertical displacement of +1, the line is y = 1.

Period: the period is the distance before the function's movement repeats itself. It can be found by $\frac{2 \pi}{|} b |$. In this case, b = 1, so the period is $2 \pi$.

Phase shift: The phase shift is the horizontal displacement. It can be found by solving the equation $x + c = 0$.
$x - \frac{\pi}{2} = 0$
$x = \frac{\pi}{2}$

Since $\frac{\pi}{2}$ is positive, the horizontal displacement is $\frac{\pi}{2}$ units right.

Now, we have enough information to graph. If there was no phase shift, you would start on the line of rotation at (0, 1). However, since there is a phase shift of $\frac{\pi}{2}$ we start at $\left(\frac{\pi}{2} , 1\right)$. Usually, you will only be asked to do one complete cycle, so that's what I'll do. In a Sine function, there are 4 parts on the x value in one cycle, so the first point can be found by dividing the period by 4: $\frac{\frac{\pi}{2}}{4} = \frac{\pi}{8}$. The point will be $\left(5 \frac{\pi}{8} , - 3\right)$, since the amplitude is 4 and the line of rotation at y = 1. The next point will be at $\left(3 \frac{\pi}{4} , 5\right)$. The final point in our cycle is $\left(2 \frac{\pi}{2} , 1\right)$, since we have finished our period.

Hopefully this helps!