# 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?

May 28, 2018

$\sin \left(x - \frac{\pi}{4}\right) - 4$

#### Explanation:

for$\text{ sine}$ or $\text{cosine}$ function always remember the general form. $a \sin \left(b x \pm c\right) \pm d \text{ /acos(bx+c)"+-d}$

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 ${45}^{0} \text{ "i.e" } \frac{\pi}{4}$ units implying negative horizontal shift(actually it is opposite of vertical shift trend)

then the function will be
$a \sin \left(b x - \frac{\pi}{4}\right) - 4$
now assuming $a = 1 , b = 1$(as it is not stated explicitly)
the function is
$\sin \left(x - \frac{\pi}{4}\right) - 4$

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