Question 1204997: The graph shows part of a sine function of the form y = A sin B(x + C) + D. Determine the values of A, B, C, and D.
https://i.ibb.co/qyG5rc1/p.jpg
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
One of the sine templates is
y = A*sin( B(x + C) ) + D
which is what your teacher gave you.
Another template is
y = A*sin( Bx + C ) + D
which is slightly different. I'll stick to the first template.
The key parameters are:
|A| = amplitude
B = used to determined the period
C = used to determine horizontal phase shift
D = used to determine vertical shift
The highest and lowest points occur when y = 7 and y = -3 respectively.
The gap between those y values is 10 units.
Count the spaces, or subtract and use absolute value, to determine this gap.
The curve is 10 units tall. Half of which is 10/2 = 5, and this is the amplitude.
It's the vertical distance from the horizontal midline to either peak or valley.
This leads to |A| = 5.
So either A = 5 or A = -5.
Either value will work. Let's go with A = 5 for simplicity.
One of the lowest points on this curve is at (0,-3)
One max point occurs at (2pi, 7)
Going from valley to neighboring peak is a horizontal distance of 2pi units.
This represents 1/2 of the full period, so the full period is 2*2pi = 4pi units.
The curve repeats itself every 4pi units along the x axis.
T = 4pi = period
T = 2pi/B
B = 2pi/T
B = (2pi)/(4pi)
B = 1/2
The parent sine function y = sin(x) maxes out when x = pi/2.
According to this graph, one max is when x = 2pi.
This is a gap of 2pi - pi/2 = 3pi/2
So what happened is that the parent sine curve had been shifted 3pi/2 units to the right.
Let's return to the idea that the max and min occur when y = 7 and y = -3 respectively.
Add those y coordinates up and divide by 2 so we can get the midpoint.
(7+(-3))/2 = 4/2 = 2
The midline is y = 2 which leads to D = 2
This is how much we shift the parent function upward.
The last thing we need is C.
Let's plug in those previous values along with the coordinates of the min (0,-3) so we can solve for C.
y = A*sin( B(x + C) ) + D
-3 = 5*sin( 0.5(0 + C) ) + 2
-3 = 5*sin(0.5C) + 2
-3-2 = 5*sin(0.5C)
-5 = 5*sin(0.5C)
sin(0.5C) = -5/5
sin(0.5C) = -1
0.5C = arcsin(-1)
0.5C = -pi/2
(1/2)C = -pi/2
C = 2*(-pi/2)
C = -pi
Answers:
A = 5
B = 1/2
C = -pi
D = 2
Confirmation using Desmos which is a graphing tool.
https://www.desmos.com/calculator/yuzs46tz03
GeoGebra is another graphing tool that I recommend.
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