Find the amplitude, period, or phase shift of the given function

Question

Find the amplitude of y = -4 cos (x + π/4)

Find the period of y = 4 cos [5x + 5π/2]

Find the period of y = 5 sin [1/2 (x - π/2)]

Find the phase shift of y = -5 cos [6 (x + π/6)]

Answer 1

Here are the 2 functions with the transformations

y = A sin(B(x + C))+D

y = A cos(B(x + C))+D

A is the amplitude

Period = 2π/B

C is the phase shift

D is the midline

For the questions

Find the amplitude of y = -4 cos (x + π/4)

Amplitude = 4

Find the period of y = 4 cos [5x + 5π/2]

Rewrite in standard form y = 4 cos [5(x + π/2)]

Period = 2π/B

Period = 2π/5

Find the period of y = 5 sin [1/2 (x - π/2)]

Period = 2π/B

Period = 2π/(1/2)

Period = 4π

Find the phase shift of y = -5 cos [6 (x + π/6)]

Phase shift = π/6

Answer 2

I'll find the amplitude, period, and phase shift for each function using the general form:

y = A cos[B(x - C)] + D or y = A sin[B(x - C)] + D

where:

1. |A| = amplitude
2. Period = 2π/|B|
3. C = phase shift (positive = right, negative = left)
4. D = vertical shift

1. Find the amplitude of y = -4 cos(x + π/4)

Here: A = -4

Amplitude = |A| = |-4| = 4

2. Find the period of y = 4 cos[5x + 5π/2]

First, rewrite in standard form by factoring out B from inside the brackets: y=4cos⁡[5(x+π2)]y=4cos[5(x+2π​)]

Here: B = 5

Period = 2π/|B| = 2π/5

3. Find the period of y = 5 sin[1/2(x - π/2)]

Here: B = 1/2

Period = 2π/|B| = 2π/(1/2) = 4π

4. Find the phase shift of y = -5 cos[6(x + π/6)]

Rewrite to identify C: y=−5cos⁡[6(x−(−π6))]y=−5cos[6(x−(−6π​))]

Here: C = -π/6

Phase shift = -π/6 (or π/6 to the left)

Summary of Answers:

1. Amplitude = 4
2. Period = 2π/5
3. Period = 4π
4. Phase shift = -π/6 (π/6 units to the left)

Answer 3

Given Information: Trigonometry Mathematical Problem

Find the amplitude of y = -4 cos (x + π/4)

Find the period of y = 4 cos (5x + 5π/2)

Find the period of y = 5 sin [1/2 (x - π/2)]

Find the phase shift of y = -5 cos [6 (x + π/6)]

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Let's put forth the effort to identify the amplitude, period and phase shift by using the necessary functions.

Step 1:

Find the amplitude

(Use a scientific calculator to enter the data to find the amplitude)

-4 cos ( x + pi divided by 4)

rewrite equation -4 times cos (pi divided by 4) = -3.999624199 roundoff = -4

Absolute value= l -4 l = 4

Answer: Amplitude = 4

Step 2:

Find the period of function:

The period of the function f(x) = A cos (B times x + C) is given by T = 2pi divided by B

Find the coefficient B of x and use the formula T= 2pi divided by B to calculate the period. T = time = period

f(x) = -4 cos ( 5x + 5 pi divided by 2)

Identify the coefficient of x

The coefficient of x in the function f (x) = -4 cos (5x + 5 pi divided by 2) B= 5

Use the formula for the period of a cosine function:

T = 2pi divided by B Substitute B = 5 into the formula: T=2pi divided by 5

Answer: The period of the function f (x) = - 4 cos (5x + 5pi/2) = 2pi divided by 5

Step 3:

Find the period of the function:

The function is f(x) = 5 sin [ ( 1/2 (x - pi divided by 2) ]

The formula for the sin function = A sin times (B times x - C) + D.

The period of the sin function is 2pi divided by B

Substituting B = 1/2 the period is 2pi divided by 1/2 = 2pi divided by 1/2 = 2pi x 2 = 4pi

Answer: The period of the function 5 sin [ ( 1/2 (x - pi divided by 2) ] = 4pi

Step 4:

Cosine function: y = A cos [B times (x - C)] + D

A = amplitude

B = used to calculate the period

C = the phase shift

D = vertical shift

Find the phase shift of y = -5 cos [6 (x + pi divided by 6)] (horizontal shift of the function)

Rewrite function as y = -5 cos [6 times (x - (-pi divided by 6) )]

Identify the phase shift using the formula y = A cos [B times (x - C)] +D, we have C = - pi divided by 6

Answer: The phase shift of the function = - pi divided by 6 direction: shifted to the left

Note: The negative sign means shifted to the left

I hope this information is useful. Please let me know if you require any more assistance. If anyone in my neighborhood is interested in setting up an in-person math tutoring session. I look forward to hearing from them. Have an amazing day. Doris H.

Answer 4

The standard form is typically written as y = Asin[B(x - C)] + D so you can think of the phase shift as the negative of the value grouped with the "x" so for:

y = -4 cos (x + π/4) the phase shift is -π/4

y = 4 cos [5x + 5π/2] first factor out the 5 to get y = 4 cos [5(x + 5π/10] then the phase shift is -5π/10 or -π/2

y = 5 sin [1/2 (x - π/2)] the phase shift is π/2

y = -5 cos [6 (x + π/6)] the phase shift is -π/6

A negative phase shift means it is shifted left and positive means it is shifted right.