Question

The sine function undergoes a vertical translation of 4 units down and a phase shift of 45 degrees to the right. what is the equation of the resulting function?

Answer 1

`sin(x-pi/4)-4`

Explanation:

for`" sine"` or `"cosine"` function always remember the general form. `asin(bx+-c)+-d " /acos(bx+c)"+-d"`

here `a` is the amplitude(or the heights above or below the origin)
`b` is the extent of shrinking or expanding
"`c` is the phase (can also be considered horizontal shift)"
"`d` is the vertical shift"

from the question ,we have vertical translation of 4 units down(implying negative vertical shift)
and phase change(or horizontal translation) is of `45^0" "i.e" "pi/4` units implying negative horizontal shift(actually it is opposite of vertical shift trend)

then the function will be
`asin(bx-pi/4)-4`
now assuming `a=1,b=1`(as it is not stated explicitly)
the function is
`sin(x-pi/4)-4`

it will be more clear from graph
graph{sin(x-pi/4)-4 [-10, 10, -5, 5]}