Vertical Translations!

Question

A sine function has an amplitude of √2, period of 8pi, horizontal shift of pi, and a vertical shift of 1.

What is the y-value of the positive function at x=4pi?

Answer 1

The function has the form y = Asin(Bx + C) + D

Vertical shift = D = 1

Amplitude = lAl = √2. Choose A = √2

Period = 2π / lBl = 8π. So, lBl = 1/4. Let B = 1/4

Horizontal shift = -C/B = -4C = π. So, C = -π/4

So, y = √2sin(x/4 - π/4) + 1

When x = 4π, y = √2sin(3π/4) + 1 = √2(1/√2) + 1 = 2

Answer 2

8pi = 2pi/|B|

8pi|B| = 2pi

|B| = 1/4

y = √2(sin(x/4 - pi)) + 1

y = √2(sin(pi - pi) + 1

y = √2(sin(0)) + 1

y = √2(0) + 1

y = 1