Answers:
- Midline: y = 0
- Amplitude = 1.5
- Function: g(x) = 1.5sin(0.5x + pi/4)
There are infinitely many possible ways to answer the third part.
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Explanation:
The midline is the horizontal line that goes through the center of the sine curve. Visually we can see that is y = 0. Another way we can see this is to note how y = 0 is the midpoint of y = 1.5 and y = -1.5, which are the max and min respectively.
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The amplitude is the vertical distance from y = 0 to y = 1.5, and it's also the vertical distance from y = 0 to y = -1.5; in short, it's the vertical distance from center to either the peak or valley.
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The general format of a sine curve is
y = A*sin(B(x-C)) + D
where,
- |A| = amplitude
- B = variable tied with the period T, specifically B = 2pi/T
- C = handles the left and right phase shift
- D = handles the vertical shifting, and y = D is the midline.
In this case, we found so far that
- |A| = 1.5 which could lead to A = 1.5
- B = 2pi/(4pi) = 0.5 since T = 4pi is the period
- D = 0
The only thing we're missing is the value of C, which is the phase shift.
Note the point (pi/2, 1.5) is one of the max points on this curve. Also, recall that sin(x) maxes out at 1 when x = pi/2
This must mean that the stuff inside the sine, the B(x-C) portion, must be equal to pi/2 in order to lead sin(B(x-C)) = 1.
So,
B(x-C) = pi/2
0.5(pi/2-C) = pi/2
pi/4 - C/2 = pi/2
4*( pi/4 - C/2 ) = 4*(pi/2)
pi - 2C = 2pi
-2C = 2pi - pi
-2C = pi
C = pi/(-2)
C = -pi/2
This allows us to update the function to get g(x) = 1.5*sin(0.5(x+pi/2)) which is the same as g(x) = 1.5sin(0.5x + pi/4)
This is one possible answer because we could have infinitely many possible values for C, due to sin(x) = 1 having infinitely many solutions.
Also, you could use cosine instead of sine. Cosine is just a phase shifted version and of sine, and vice versa.
