Question

A humble request for help.

Answer 1

`y=1.9 \cos\left((2\pi)/(5.6)(x-0.3)\right)+2.3`

Explanation:

The given formula of a cosine function is:

`y=A\cos \left((2\pi)/(T)(x-C)\right)+D`

where:

• A is the amplitude (vertical height from the midline to the peak).
• T is the period (horizontal distance between consecutive peaks).
• C is the phase shift (horizontal shift - negative is to the right).
• D is the vertical shift.

From observation of the given graph:

• The y-values of the peaks are y = 4.2.
• The y-values of the troughs are y = 0.4.

The amplitude (A) is half the distance between the y-values of the peaks and troughs. Therefore:

`A=(4.2-0.4)/(2)=1.9`

The period (T) of the function is the difference between the x-values of two consecutive peaks. Therefore:

`T = 5.9 - 0.3 = 5.6`

The parent cosine function has a maximum point when x = 0. Therefore, since the x-value of the peak of the graphed function is x = 0.3, the graphed function has been shifted horizontally to the right by 0.3 units. This means that we need to subtract 0.3 from the x-variable, so:

`C = 0.3`

The midline is the midpoint of the y-values of the minimum and maximum points. Therefore:

`\textsf{Midline:}\quad y = (4.2+0.4)/(2)=2.3`

As the midline of the parent cosine function is the x-axis (y = 0), the graphed function has been shifted up 2.3 units. Therefore:

`D = 2.3`

So, the formula for the graphed function is:

`\Large\boxed{\boxed{y=1.9 \cos\left((2\pi)/(5.6)(x-0.3)\right)+2.3}}`