Question 1205033
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The highest and lowest points occur when y = 4 and y = -4 respectively.
Use the midpoint formula to find that y = 0 is the midpoint. 
This leads us to <font color=red>D = 0</font> which represents the midline.
This applies to both sine and cosine templates.


The height of this curve is 8 units. 
Half of which is 4, which is the value of A.
<font color=red>A = 4</font>
This applies to both sine and cosine templates.
Technically we could go for A = -4, but I'll stick with the positive version since that's what your teacher picked.


The neighboring peak points occur when x = 0.5 and x = 2.5
The gap is 2.5 - 0.5 = 2 units which is the period.
The curve repeats itself every 2 units along the x axis.


T = period = 2
T = 2pi/B
B = 2pi/T
B = 2pi/2
<font color=red>B = pi</font>
This applies to both sine and cosine templates.


So far we found that:
<font color=red>A = 4</font>
<font color=red>B = pi</font>
<font color=red>D = 0</font>
and they work for both sine and cosine templates mentioned.
Side note: Cosine is a phase shifted version of sine. 


Let's plug those values in. We'll also plug in one of the points on the curve.
I'll plug in (0.5, 4)
This will allow us to solve for C.


So,
y = A*sin( B(x+C) ) + D
4 = 4*sin( pi(0.5+C) ) + 0
4 = 4*sin( pi(0.5+C) )
sin( pi(0.5+C) ) = 4/4
sin( pi(0.5+C) ) = 1
pi(0.5+C) = arcsin(1)
pi(0.5+C) = pi/2
0.5+C = 1/2
1/2+C = 1/2
C = 1/2 - 1/2
<font color=red>C = 0</font>


Summary for the sine template
<font color=red>A = 4</font>
<font color=red>B = pi</font>
<font color=red>C = 0</font>
<font color=red>D = 0</font>
This will mean we go from y = A*sin(B(t+C)) + D to <font color=red>y = 4*sin(pi*t)</font>


Cosine will look almost identical in terms of A,B,C,D values. 
A,B,D will be the same.
C is going to be different.


Let's plug the known A,B,D values into the cosine template. 
Also let's plug in (0.5, 4) so we can solve for C.
y = A*cos(B(x+C)) + D
4 = 4*cos(pi(0.5+C)) + 0
4 = 4*cos(pi(0.5+C))
cos(pi(0.5+C)) = 1
pi(0.5+C) = arccos(1)
pi(0.5+C) = 0
0.5+C = 0
C = -0.5
<font color=red>C = -1/2</font>


For cosine we have
<font color=red>A = 4</font>
<font color=red>B = pi</font>
<font color=red>C = -1/2</font>
<font color=red>D = 0</font>
which explains how we arrive at <font color=red>y = 4*cos(pi(t - 1/2))</font>


Confirmation using Desmos graph
<a href="https://www.desmos.com/calculator/uatjjocx1m">https://www.desmos.com/calculator/uatjjocx1m</a>
I'm using x in place of t much of the time, but you get the idea.
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