Question 968758
the general formula for the sine function 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.


the normal period of the sine function is 2*pi.


the formula for the period is:


period = (2 * pi) / freqauency.


when a or b or c or d are not shown, their default values are:
a = 1
b = 1
c = 0
d = 0


when d = 0, the center line of the graph is at y = 0.
when d = -1, the center line of the graph is at y = -1.


when b = 1, the period = 2 * pi divided by 1 which is equal to 2 * pi.



your formula brecomes:


f(t) = 4 * sin(pi/2 * x) - 1


your amplitude = 4
your frequency = pi/2
your horizontal shift = 0
your vertical shift = -1



the center line of your graph is at y = -1
the maximum value of your graph is -1 + 4 = 3
the minimum value of your graph is -1 - 4 = -5


your period is equal to 2 * pi divided by pi/2.
this makes your period equal to 5.


your horizontal shift is one unit to the right.
this one is a litle tricky to figure out, so we'll go through a calculation to show you what happens.


your questions were:


a) What is the amplitude of f(t)? 


the amplitude is equal to 4


(b) What is the period of f(t)?


the period is 4 radians



(c) What are the maximum and minimum values attained by f(t)? 


the maximum value is f(T) = 3
the minimum value is f(t) = -5


(d) Sketch the graph of f (t) for t ∈ [−1, 3].


the graph is shown below:


<img src = "http://theo.x10hosting.com/2015/050602.jpg" alt="$$$" </>


the period from x = -1 to x = 3 is marked between vertical dashed lines.


the center line of the graph is at y = -1


the maximum value is at y = 3 and the minimum value is at y = -5.


you can see that the graph crosses the center line at x = -1, x = 1, x = 3 and x = 5.


the center line is at y = -1.


take away the vertical displacement of -1 and the center line would have been at the x-axis.


you would analyze this graph as follows:


when x = 3, the equation of the graph becomes:


y = 4 * sin(pi/2 * (3-1)) - 1


this becomes y = 4 * sin(pi/2 * 2) - 1


this becomes y = 4 * sin(pi) - 1


sin(pi) is equal to 0.


the equation becomes y = 4 * 0 - 1


it finally becomes y = -1.


you can see on the graph that, when x = 3, y = -1.


this agrees with the equation.


we'll do one more.


when x = -1, the equation of y = 4 * sin(pi/2 * (x-1)) - 1 becomes:


y = 4 * sin(pi/2 * (-1-1)) - 1 which becomes:


y = 4 * sin(pi/2 * -2) - 1 which becomes:


y = 4 * sin(-pi) - 1 which becomes:


y = 4 * 0 - 1 which becomes:


y = -1.


when x = -1, the value of y is equal to -1.


the graph confirms that.