Question

What is the midline of this function? y=    

Answer 1

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.