Write the equation of the transformed graph of sine with period that has been shifted vertically down 4 units and has an amplitude of 2. (3 points)

Question

Write the equation of the transformed graph of sine with period that has been shifted vertically down 4 units and has an amplitude of 2. (3 points)

Answer 1

y = ±a sin (bx – c) + d

The fourth coefficient d is for the vertical shift

a = amplitude = 2

There is no flipping across the x-axis means start with a = 2

period = π = 2π/b, so b = 2π/(period) = 2π/π = 2

there’s no phase shift, so c = 0

shifted vertically down 4 units means d = –4

So our required equation is

y = 2 sin (2x) – 4

Answer 2

y = 2sin(2x) -4

The 2 in front of the sine function produces an amplitude change from 1 to b 2.

The -4 shifts the entire function down 2 units.

The normal period of a sine wave is 2π. To change that to π, we need to double the frequency. And that is exactly what 2x does as an altered input to the sine function.

https://www.desmos.com/calculator/suhwb3v7lh

Answer 3

The general formula for these transformations is

y = A sin b (x - h) + k

The A is the amplitude

b influences the period. Use it as follows:

new period = 2π / b.

h is the horizontal shift.

Take note that (x - π) shifts π units to the RIGHT, while (x + π ) shifts π units to the LEFT.

k shifts the graph vertically.

Example: y = 3 sin 2 (x - π/4) + 5

This is the sine wave with an amplitude of 3, a period of 2π / 2. or π ,

shifted π/4 to the RIGHT and 5 units UP

There is a wonderful video on YouTube at

https://www.youtube.com/watch?v=U3ktjhKRdng

Note that the cosine function is done similarly. With tangent, the new period is calculated using just π / b.