i wanted to make sure that both functions are the same.
y = 2 sin (4x + pi) +3
=
y = 2 sin 4 ((x) + (pi/4)) + 3
sin ( theta + pi ) = - sin ( theta )
sin ( 4x + pi ) = - sin ( 4x )
2 sin ( 4x + pi ) = - 2 sin ( 4x )
2 sin (4x + pi) + 3 = - 2 sin (4x) + 3 =
3 - 2 sin (4x)
For the second:
y = 2 sin 4 ((x) + (pi/4)) + 3
= 2 sin (x + pi) + 3 , I just distributed the 4 over the bracket
so, yes, they are the same
To determine if both functions are the same, we need to simplify and compare them.
First, let's simplify each function:
Function 1:
y = 2 sin (4x + pi) + 3
Function 2:
y = 2 sin 4 ((x) + (pi/4)) + 3
In Function 1, we have sin (4x + pi). This means the function has a sine wave with a period of 2*pi/4 = pi/2 (since the coefficient of x is 4).
In Function 2, we have sin 4 ((x) + (pi/4)). This means the function has a sine wave with a period of 2*pi/4 = pi/2.
So, both functions have the same period of pi/2.
Now let's compare the phase shifts:
In Function 1, the phase shift is -pi/4. This means the sine wave is shifted to the right by pi/4 units.
In Function 2, the phase shift is pi/4. This means the sine wave is shifted to the left by pi/4 units.
Since the phase shifts have opposite signs, the functions are not the same.
To verify this, we can graph both functions on a graphing calculator or software and observe the differences in their shapes and positions.