Question

Give an equation for a transformed sine function with an amplitude of 1/2, 1 a period of 5/2' a phase shift of 3/2 rad to the right, and a vertical translation of 1 unit down

Answer 1

To construct the required transformed sine function with an amplitude of 1/2, a period of 5/2, a phase shift of 3/2 rad, and a vertical translation of 1 unit down, use the equation y = (1/2) sin[(4π / 5)(x - 3/2)] - 1.

Step-by-step explanation:

To construct a transformed sine function that meets the specified criteria, we start with the basic sine function and apply transformations for amplitude, period, phase shift, and vertical translation. The general form of a sine function is y = A sin(B(x - C)) + D, where

• A is the amplitude
• B affects the period (Period = 2π / B)
• C is the phase shift (to the right is positive)
• D is the vertical translation

We are given an amplitude of 1/2, a period of 5/2, a phase shift of 3/2 rad to the right, and a vertical translation of 1 unit down. To apply these to our equation:

• The amplitude (A) is 1/2.
• For a period of 5/2, we calculate B as B = 2π / (5/2) = 4π / 5.
• The phase shift (C) is 3/2 rad, so we use this directly in the equation.
• A vertical translation of 1 unit down signifies D = -1.

Putting it all together, we get the equation:

y = (1/2) sin[(4π / 5)(x - 3/2)] - 1

Answer 2

The equation for the transformed sine function is:

y = 0.5 sin (2π/5 * (x - 3/2)) - 1

Step-by-step explanation:

Amplitude: The amplitude is adjusted by multiplying the sine function by 1/2.

Period: The period is stretched by a factor of 5/2 by dividing the argument by 2π/5.

Phase shift: The phase shift of 3/2 units to the right is achieved by adding 3/2 to the argument of the sine function.

Vertical translation: The vertical translation of 1 unit down is achieved by subtracting 1 from the entire expression.

Combining these adjustments, we obtain the final equation:

y = 0.5 sin (2π/5 * (x - 3/2)) - 1