Question

Write a cosine function that has an amplitude of 2, a midline of 5 and a period of T.

Answer 1

phase shift is 0

General equation

• y=AcosB(x-C)+D

c is phase shift

• y=AcosBx+D

Midline is D

• y=AcosBx+5

A is amplitude

• y=2cosBx+5

B is period

• 2π/T

So

Final equation

• y=2cos(2πx/T)+5

Not mandatory from now

For some special cases 2π/T=omega

So Equation yields

• y=2cos(
  `\omega x`)+5

Answer 2

`f(x)=2 \cos (2x)+5`

Explanation:

The cosine function is periodic, meaning it repeats forever.

Standard form of a cosine function:

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

• A = amplitude (height from the mid-line to the peak)
• 2π/B = period (horizontal distance between consecutive peaks)
• C = phase shift (horizontal shift - positive is to the left)
• D = vertical shift

Given:

• Amplitude = 2 ⇒ A = 2
• 
  `\sf Period=\pi \implies (2 \pi)/(B)=\pi \implies B=2`
• mid-line = 5 ⇒ D = 5

Inputting the given values into the standard form:

`\implies f(x)=2 \cos (2x)+5`