Question 920439
the general formula is:


y = a*(sin(b(x-c))+d


|a| is the amplitude
b is the frequency
c is the horizontal shift
d is the vertical shift.


replace t with x and your equation is:


y = -2 * sin( pi * x + pi/3)


this can also be written as:


y = -2 * sin( pi * (x + 1/3))


the general equation is, once again:


y =  a * (sin( b * (x-c)) + d


in your equation:


|a| = 2
b = pi
c = 1/3
d = 0


your amplitude is equal to 2
your frequency is equal to pi
your horizontal shift is equal to -1/3
your vertical shift is equal to 0


your period is equal to 2*pi divided by your frequency.
this makes your period equal to 2*pi / pi = 2


everything is in radians.


your equation looks like this:


<img src = "http://theo.x10hosting.com/2014/1105_0002.jpg" alt="picture 1" </>


the period is equal to 2 and goes from -1/3 to 5/3.
your amplitude is equal to 2 so the graph goes from a high of 2 to a low of -2.
there is no vertical shift because d = 0.


take away the horizontal shift and your graph looks like this:


<img src = "http://theo.x10hosting.com/2014/1105_0004.jpg" alt="picture 2" </>


everything else remains the same except there is no more horizontal shift so the period goes from 0 to 2.


your y intercept is the value of y when x = 0.


your equation is, once again:


y = -2 * sin( pi * (x + 1/3))


when x = 0, y = -2 * sin(pi/3)


pi/3 = 60 degrees and sin (60) = sqrt(3)/2


-2*sqrt(3)/2) = -sqrt(3) which is equivalent to -1.73.....


you can see that on the following graph.


that's the value of y when x = 0


<img src = "http://theo.x10hosting.com/2014/1105_0005.jpg" alt="picture 3" </>


the x-intercept is the value of x when y = 0


set your equation and solve for x.


you get:


0 = -2sin(pi(x+1/3))


divide both sides by -2 and you get:


sin(pi(x+1/3) = 0


this is true if and only if arc-sine(0) = pi(x+1/3)


arc-sin(0) = 0 so pi(x+1/3) = 0 which results in x = -1/3.


the value of y is 0 when x = -1/3.


<img src = "http://theo.x10hosting.com/2014/1105_0006.jpg" alt="picture 4" </>


here's a good reference on the general formula used.


<a href = "http://www.regentsprep.org/regents/math/algtrig/ATT7/sinusoidal.htm" target = "_blank">http://www.regentsprep.org/regents/math/algtrig/ATT7/sinusoidal.htm</a>