SOLUTION: The depth of the water at the end of a boat dock is given by: y=3sin[2pi/12 (x+3)] +9, where x is the number of hours after midnight. a. What is the amplitude? b. What is th

Question 607375: The depth of the water at the end of a boat dock is given by:
y=3sin[2pi/12 (x+3)] +9, where x is the number of hours after midnight.
a. What is the amplitude?
b. What is the period?
c. What is the phase shift?
d. Sketch the graph of function for one full period starting with midnight.
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
The depth of the water at the end of a boat dock is given by:
y=3sin[2pi/12 (x+3)] +9, where x is the number of hours after midnight.
a. What is the amplitude?
b. What is the period?
c. What is the phase shift?
d. Sketch the graph of function for one full period starting with midnight
**
Equation of sin function: Asin(Bx-C), A=amplitude, Period=2π/B, Phase shift=C/B
Rewrite given equation:
y=3sin[2pi/12 (x+3)] +9
=3sin[π/6 (x+3)] +9
=3sin[πx/6+π/2] +9
..
Amplitude=3
B=π/6
Period=2π/B=2π/(π/6)=12 hrs
1/4 period=3
C=π/2
Phase shift=C/B=(π/2)/(π/6)=3 hrs (shift left)
..
Graph function for one period:
I don't have the means to draw a graph but I will develop the (x,y) coordinates with which you can use to draw a graph for one period.
..
Start with coordinates of basic sin function: sin(πx/6) (x-axis scaled in hours)
(0,0), (3,1), (6,0), (9,-1), (12,0)
Include amplitude: 3sin(πx/6)
(0,0), (3,3), (6,0), (9,-3), (12,0)
Shift 3 hrs left: 3sin(πx/6+π/2)
(-3,0), (0,3), (3,0), (6,-3), (9,0),(12,3)
Bump curve up 9 units:3sin(πx/6+π/2)+9 (final configuration)
(-3,9), (0,12), (3,9), (6,6), (9,9), (12,12)
..
y-intercept
set x=0
y=3sin(πx/6+π/2)+9
y=3(sinπ/2)+9
y=3*1+9=12
You now have the y-intercept and the (x,y) coordinates with which you can draw graph of given sin function. One full period starts at (0,12) and ends at (12,12)

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