In trigonometric functions, is horizontal translation affected by y-axis reflection?

Question

f(x) = a( sin bx +c) +d

f(x) = -(sin(x-pi/2))

b = 1, c = -pi/2

horizontal translation is -c/|b| = -(-pi/2)/|1| = pi/4

so,

the horizontal translation is pi/4, is it to the right or to the left?

I know it's positive but a = -1, so there's a y-axis reflection

does that mean the horizontal translation is now to the left?

and if that's the case (y-axis reflection affects horizontal translation), then is this applied in all functions? e.g. logarithmic?

and does x-axis reflection affect vertical translation?

Answer 1

It is moved right horizontally by pi/4 units. The a=-1 reflection is over the x axis, therefore only y values are changed, x-values remain the same. Just try different combinations on your graphing calculator or on Desmos and you will see what happens.