# How do you graph y=3 sin (2x-pi) + 2?

Dec 17, 2017

graph{y=3sin(2x-pi)+2 [-10, 10, -5, 5]} Use the rules of graphing sine equations.

#### Explanation:

The things that you need to find are the amplitude, the period length, the phase shift, and the amount that it is shifted up or down. The amplitude is represented as A. The period length is the general period of $2 \pi$ divided by the number represented by B. The shift upwards or downwards is C, and the phase shift is D. So, a general sine equation looks like this:
$y = C + A \sin B \left(x + D\right)$

However, C can be and commonly is moved to the end, making it look like this:
$y = A \sin B \left(x - D\right) + C$
This does not change the application of C.

So, to apply this knowledge, you have to understand what each of those means. The amplitude is how far off of the midline of the graph it rises and falls. Technically, it's the absolute value of A, but whether A is positive or negative determines whether you go down first or up first. So, in this case, your amplitude is 3.

The phase shift is how far to the left or right you move the graph. As it is represented by $\left(x - D\right)$, it is important to remember that addition in the parentheses means that it is actually a negative number, and vice versa. It is also important to note that x must not have a coefficient in the parentheses, and you will have to divide it out if there is one. So, in this case, you would first need to divide $\frac{2 x - \pi}{2}$, which equals $2 \left(x - \frac{\pi}{2}\right)$. That 2 is your B. So, your phase shift equals $\frac{\pi}{2}$.

The period length is how far one period, or one sine wave, goes. A natural period length is $2 \pi$, and to find any other period length, you divide $\frac{2 \pi}{B}$. Since when we established the phase shift, we also established that $B = 2$, we can now figure out the phase shift. $\frac{2 \pi}{2} = \pi$ So, your period length is $\pi$.

The vertical shift moves the midline of the graph up or down, depending on whether it is positive or negative. In this case, your $C = 2$, so you would move the midline of the graph up two units.