Write a Sine Function

Question

Write a sine function that has a midline of y= -4, an amplitude of 4 and a period of 7.

Answer 1

Consider the standard form of the sine function:

1. y = Asin⁡(B(x − C)) + D

where:

1. A is the amplitude
2. B affects the period of the function
3. C is the horizontal shift (phase shift)
4. D is the vertical shift (midline)

Given:

1. Amplitude (A) = 4
2. Midline (D) = -4
3. Period = 7

The period of a sine function is given by 2π / B​. Therefore, to find B:

1. Period = 2π / B
2. 7 = 2π / B
3. B = 2π / 7

There is no horizontal shift given, so C = 0.

Substitute the values into the sine function form:

1. y = 4sin⁡((2π/7) * x) − 4

Hope this was helpful.

Answer 2

The sine function that has a midline of y= -4, an amplitude of 4 and a period of 7 can be written as y = 4 sin (2πx/7) -4

The amplitude=a=4

The period of 7 = 2π/b ⇒ b=2π/7

The midline is c=-4

So,

y = 4 sin((2π / 7)(x)) - 4

Explanation:

The general form of a sine function is y = a sin(bx) + c

Amplitude = |a| 

1. This is how tall the wave is by measuring the length of the crest (peak point of sine wave) to the midline.

Period = 2π/|b|

1. This is how long one cycle is

The midline = c. 

1. If y=a sin x → [-a,a] would be the vertical distance between the peaks and the troughs.
2. If y = sin x + c → [-a+c,a+c] would be the vertical distance between the peaks and the troughs.

For example, for y=sin(x):

1. a=1, b=1, c=0
2. the midline is y=0, the amplitude is 1, and period is 2π.