Write the equation of the cosine function with an amplitude of 1, a period of 3pi/7, a phase shift of -pi/5, and a vertical shift of 2 units down.

The only actual calculation you have to do deals witht the period.

period = 2π/k
3π/7 = 2π/k
k =

I am also not clear about your information re phase shift,
I will assume that by " a phase shift of -π/5" you mean a shift of π/5 to the left.
If you meant the opposite, just change the sign for the phase shift in my answer.

y = cos(14/3)(Ø + π/5) - 2

i didn't mean to put a negative sign

To write the equation of the cosine function with the given characteristics, we can use the general form of the cosine function:

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

Where:
A = amplitude
B = 2π / period
C = phase shift
D = vertical shift

In this case:
Amplitude (A) = 1
Period = 3π/7
Phase shift (C) = -π/5
Vertical shift (D) = -2 (2 units down)

Now, let's substitute these values into the equation:

y = 1 * cos(2π / (3π/7))(x - (-π/5)) - 2

Simplifying further:

y = cos((14/3π)(x + π/5)) - 2

So, the equation of the cosine function with an amplitude of 1, a period of 3π/7, a phase shift of -π/5, and a vertical shift of 2 units down is y = cos((14/3π)(x + π/5)) - 2.

To write the equation of the cosine function with the given characteristics, we can use the general form of the cosine function:

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

where:
A represents the amplitude
B represents the period
C represents the phase shift
D represents the vertical shift

In this case, we are given:
Amplitude (A) = 1
Period (B) = 3π/7
Phase shift (C) = -π/5
Vertical shift (D) = 2 units down

Therefore, the equation of the cosine function is:

y = 1 * cos((2π/period)(x - phase shift)) + vertical shift

Substituting the given values:

y = 1 * cos((2π/(3π/7))(x - (-π/5))) + 2

To simplify further:

y = 1 * cos((2π/(3π/7))(x + π/5)) + 2

Now, we can simplify the expression inside the cosine function:

y = 1 * cos((2π * 7/3π)(x + π/5)) + 2

Simplifying:

y = 1 * cos((14/3)(x + π/5)) + 2

Thus, the equation of the cosine function with an amplitude of 1, a period of 3π/7, a phase shift of -π/5, and a vertical shift of 2 units down is:

y = cos((14/3)(x + π/5)) + 2