Question

NO LINKS!! Write the equation of the trigonometric graph. Try fractional values or π for the box next to x.​

Answer 1

`y=\boxed{1}\: \cos(\:\boxed{1}\:x)+\boxed{3}`

Explanation:

General form of cos periodic function:

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

where:

• A = Amplitude (height from the center line to the peak or trough)
• 2π/B = Period (horizontal distance between one peak and the next)
• C = (horizontal) Phase Shift
• D = Vertical Shift

Parent function → y = cos(x)

The parent function y = cos(x) has a center line at y = 0.

The center line of the new function is at y = 3, so the parent function has been shifted vertically by 3 units. Therefore, D = 3:

⇒ y = A cos(B(x + C)) + 3

The amplitude of the parent function is 1.

The amplitude of the new function is also 1. Therefore, A = 1:

⇒ y = 1 cos(B(x + C)) + 3

The parent function has a peak at x = 0.

The new function has a peak at x = 0. Therefore there has been no horizontal phase shift, so C = 0:

⇒ y = 1 cos(B(x + 0)) + 3

⇒ y = 1 cos(Bx) + 3

Finally, from inspection of the curve, the period appears to be 2π (6.28)

2π/B = 2π so B = 1

⇒ y = 1 cos(1x) + 3

Please see attached graph for reference.

• The parent function is shown in blue.
• The dotted line is shown in grey.
• The new function is shown in black.

Answer 2

y = 1·cos(π/3·x) +3

Explanation:

The general form of the trig equation will be ...

y = (amplitude)·cos(2π/(period)·x) +(midline shift)

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midline

You have already determined correctly that the midline shift is 3 units.

amplitude

The amplitude is the difference between the peak value (4) and the midline, so is ...

amplitude = peak - midline = 4 -3 = 1

period

The period is the horizontal distance between corresponding parts of the graph. Here, the distance between peak values is 6 units, so the period is 6. That means the argument of the cosine function is ...

(2π)/6·x = (π/3)·x

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The function you are looking for is ...

`y=1\cdot\cos\left((\pi)/(3)x\right)+3`