Write an equation of the tangent function with period (3π)/8, phase shift -π/5, and vertical shift -2.

a. y=tan((8x)/3 - (8π)/15) -2
b. y=tan((8x)/3 + (8π)/15) -2
c. y=tan((16x)/3 +(3π)/80) -2
d. y=tan((8x)/3 - (3π)/40) -2

I think it is A... or B?

personally, I prefer to write trig functions with the k factor outside the bracket. That way the sphase shift is immediately obvious.

so my first choice would have been

y = tan (8/3)(x + π/5) - 2

which after expanding, is the same as B

thanks

The correct answer is B.

To write the equation of the tangent function with a period of (3π)/8, phase shift of -π/5, and vertical shift of -2, we use the general form of the tangent function:

y = A * tan(Bx - C) + D

Where A determines the amplitude, B determines the period, C determines the phase shift, and D determines the vertical shift.

In this case, the period is (3π)/8, so B = 2π/((3π)/8) = 16/3.
The phase shift is -π/5, so C = π/5.
The vertical shift is -2, so D = -2.

Plugging in these values, we get:

y = tan((8x)/3 + (8π)/15) - 2

Therefore, option B is correct.

To determine the correct equation of the tangent function with the given period, phase shift, and vertical shift, we can use the general form of the tangent function:

y = A * tan(Bx - C) + D

where:
A represents the amplitude, which is not given in the question.
B represents the horizontal stretching or shrinking factor, which is determined by the period.
C represents the phase shift.
D represents the vertical shift.

Given:
Period = (3π)/8
Phase shift = -π/5
Vertical shift = -2

1. Determine the amplitude (A):
The amplitude of the tangent function is not given in the question, which means it remains at its default value of 1.

2. Determine the horizontal stretching or shrinking factor (B) from the period:
The period of the tangent function is (3π)/8, which means one full cycle occurs over this interval. The general formula for the period of the tangent function is π/B. So, we can set up the equation:

(3π)/8 = π/B

To solve for B, cross-multiply and simplify:

3B = 8
B = 8/3

3. Determine the phase shift (C):
The phase shift is given as -π/5. Since C in the general equation is the opposite sign of the given phase shift, we have:

C = π/5

4. Determine the vertical shift (D):
The vertical shift is given as -2. So, D = -2.

Substitute the obtained values of A, B, C, and D into the general equation:

y = 1 * tan((8/3)x - (π/5)) - 2

Simplifying the equation further:

y = tan((8/3)x - π/5) - 2

Comparing this simplified equation with the given options, we can conclude that the correct answer is option B:

y = tan((8x)/3 + (8π)/15) - 2.