Write an equation of secant with period = pi, phase shift = -pi/2, and vertical shift = 3

To write the equation of the secant function with the given period, phase shift, and vertical shift, we can start with the standard equation of the secant function:

y = A sec(Bx - C) + D

where A represents the amplitude (which is not specified), B represents the frequency, C represents the phase shift, and D represents the vertical shift.

Given:
Period = π
Phase shift = -π/2
Vertical shift = 3

First, let's find the frequency, which is the reciprocal of the period:
Frequency = 1 / Period = 1 / π

Next, let's substitute the given values into the equation:
y = A sec(Bx - C) + D

Vertical shift = 3
y = A sec(Bx - C) + 3

Phase shift = -π/2
Bx - C = -π/2
Bx = -π/2 + C

Setting Bx = 0 for the phase shift, we can solve for C:
Bx = 0
0 = -π/2 + C
C = π/2

Now we have:
y = A sec(Bx - C) + 3
y = A sec(Bx - π/2) + 3

Finally, we can substitute the frequency (1/π) and simplify the equation:
y = A sec((1/π)x - π/2) + 3

This is the equation of the secant function with the given period, phase shift, and vertical shift. The value of A (amplitude) is not specified, so it could be any non-zero real number.