Question

A given sinusoidal function has a period of 3, an amplitude of 7, and a maximum at (0,2). Represent the function with a sine equation and a cosine equation.

Answer 1

To represent the given sinusoidal function with a sine equation, use the equation y = A sin(Bx + C) + D. For this particular function, the sine equation would be y = 7 sin((2π/3)x) + 2.

To represent the function with a cosine equation, use the same equation, but with a phase shift of π/2.

Step-by-step explanation:

To represent the given sinusoidal function with a sine equation, we need to use the equation:

• y = A sin(Bx + C) + D

where:

• A is the amplitude
• B determines the period (T) of the function, where T = 2π/B
• C represents any phase shift
• D is the vertical shift (if applicable)

Based on the given information, we can determine the values for A, B, C, and D:

• Amplitude (A) = 7
• Period (T) = 3, which means B = 2π/3
• Maximum is at (0,2), so there is no phase shift, C = 0
• The maximum occurs at (0,2), which means that the vertical shift is D = 2.

Putting all these values into the equation, we get:

• y = 7 sin((2π/3)x) + 2

To represent the function with a cosine equation, we can use the same equation, but with a phase shift of π/2:

• cos(x) = sin(x + π/2)
• y = 7 cos(2πx/3 - π/2) + 2

Answer 2

Step-by-step explanation:

The general form of a sinusoidal function is:

y = A sin(Bx + C) + D

where A is the amplitude, B determines the period (B = 2π/period), C is the phase shift (horizontal shift), and D is the vertical shift.

From the given information, we know that the period T = 3 and the amplitude A = 7. The maximum occurs at (0,2), which means that the vertical shift is D = 2.

Using the amplitude and vertical shift, we can write the equation in the form:

y = 7 sin(Bx) + 2

To find B, we can use the period formula:

T = 2π/B

Substituting T = 3, we get:

3 = 2π/B

B = 2π/3

Therefore, the sine equation for the given function is:

y = 7 sin(2πx/3) + 2

To find the cosine equation, we can use the identity:

cos(x) = sin(x + π/2)

Therefore, the cosine equation for the given function is:

y = 7 cos(2πx/3 - π/2) + 2