Question

What is the amplitude, period, and phase shift of f(x) = −4 sin(2x + π) − 5?

Amplitude = −4; period = 2π; phase shift: x = -pie/2
Amplitude = −4; period = π; phase shift: x = pie/2
Amplitude = 4; period = π; phase shift: x = -pie/2
Amplitude = 4; period = 2π; phase shift: x = pie/2

Answer 1

Amplitude = -4; period = π; phase shift: x = π/2

Explanation:

* Lets revise the trigonometry translation

- If the equation is y = A sin (B(x + C)) + D

* A is the amplitude

- The amplitude is the height from highest to lowest points and

divide the answer by 2

* The period is 2π/B

- The period is the distance from one peak to the next peak

* C is the horizontal shift (phase shift)

- The horizontal shift is how far the function is shifted to left

(C is positive) or to right (C is negative) from the original position.

* D is the vertical shift

- The vertical shift is how far the function is shifted vertically up

(D is positive) or down (D is negative) from the original position.

* Now lets solve the problem

∵ f(x) = A sin (B(x + C)) + D

∵ f(x) = -4 sin (2x + π) - 5 ⇒ take 2 from the bract (2x + π) common factor

∴ f(x) = -4 sin 2(x + π/2) - 5

∴ A = 4 , B = 2 , C = π/2 , D = -5

∵ A is the amplitude

∴ The amplitude is -4

∵ The period is 2π/B

∴ The period = 2π/2 = π

∵ C is the horizontal shift (phase shift)

∴ The phase shift π/2 (to the left)

* Amplitude = -4; period = π; phase shift: x = π/2