Question

Plsss help me w/ this asap !!

Answer 1

`\sf y=3.3\cos \left((2\pi)/(12.2)\left(x-0.6\right)\right)+3.9`

Explanation:

The provided general formula of the graphed cosine function is:

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

The midline of a cosine function is the horizontal line located at a y-value that is midway between the maximum (peak) and minimum (trough) y-values of the function.

From observation of the given graph:

• The maximum y-values of the peaks are y = 7.2.
• The minimum y-values of the troughs are y = 0.6.

Therefore, the midline is:

`\sf y=(7.2+0.6)/(2)=(7.8)/(2)=3.9`

Since the midline of the parent cosine function y = cos(x) is the x-axis (y = 0), the graphed function has been shifted up by 3.9 units, resulting in a vertical shift (D) of 3.9.

`\sf D=3.9`

The amplitude (A) of a cosine function is the vertical distance between the midline and the y-value of the maximum point (peak). Therefore:

`\sf A=7.2-3.9=3.3`

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

`\sf T=12.8-0.6=12.2`

The parent cosine function y = cos(x) has a maximum point when x = 0. Therefore, since the x-value of the peak of the graphed function is x = 0.6, the graphed function has been shifted horizontally to the right by 0.6 units. This means that we need to subtract 0.6 from the x-variable to account for the horizontal shift so:

`\sf C = 0.6`

So, the formula for the graphed cosine function is:

`\sf y=3.3\cos \left((2\pi)/(12.2)\left(x-0.6\right)\right)+3.9`