Certainly! Let's determine the phase shift for each of the sinusoidal curves given.
### Part (a):
The equation given is:
[tex]\[ y = 3 \sin \left(5\left(x - \frac{\pi}{4}\right)\right) + 2 \][/tex]
To find the phase shift, let's consider the general form of a sine function:
[tex]\[ y = a \sin(b(x - c)) + d \][/tex]
where:
- [tex]\( a \)[/tex] is the amplitude (vertical stretch/compression),
- [tex]\( b \)[/tex] is the frequency (horizontal stretch/compression),
- [tex]\( c \)[/tex] is the phase shift (horizontal translation),
- [tex]\( d \)[/tex] is the vertical shift.
In the given equation [tex]\( y = 3 \sin \left(5\left(x - \frac{\pi}{4}\right)\right) + 2 \)[/tex]:
- The coefficient [tex]\( a = 3 \)[/tex] (this represents the amplitude but doesn't affect the phase shift),
- The frequency [tex]\( b = 5 \)[/tex],
- The phase shift [tex]\( c = \frac{\pi}{4} \)[/tex],
- The vertical shift [tex]\( d = 2 \)[/tex].
The phase shift for a sine function is determined by the term inside the parenthesis, [tex]\( x - c \)[/tex]. Here, [tex]\( c = \frac{\pi}{4} \)[/tex]. Thus, the phase shift is:
[tex]\[ \frac{\pi}{4} \quad \text{to the right} \][/tex]
### Part (b):
The equation given is:
[tex]\[ y = -2 \cos \left(\frac{1}{3}(x + 8)\right) + 1 \][/tex]
To find the phase shift, let's consider the general form of a cosine function:
[tex]\[ y = a \cos(b(x - c)) + d \][/tex]
where:
- [tex]\( a \)[/tex] is the amplitude (vertical stretch/compression),
- [tex]\( b \)[/tex] is the frequency (horizontal stretch/compression),
- [tex]\( c \)[/tex] is the phase shift (horizontal translation),
- [tex]\( d \)[/tex] is the vertical shift.
In the given equation [tex]\( y = -2 \cos \left(\frac{1}{3}(x + 8)\right) + 1 \)[/tex]:
- The coefficient [tex]\( a = -2 \)[/tex] (this represents the amplitude but doesn't affect the phase shift),
- The frequency [tex]\( b = \frac{1}{3} \)[/tex],
- The phase shift can be tricky here, as the expression inside the cosine is [tex]\( b(x + 8) \)[/tex], we need to rewrite it to compare it with the standard form [tex]\( b(x - c) \)[/tex].
We can rewrite:
[tex]\[ \frac{1}{3} (x + 8) = \frac{1}{3} x + \frac{1}{3} \cdot 8 \][/tex]
which simplifies to:
[tex]\[ \frac{1}{3} x + \frac{8}{3} \][/tex]
Therefore, we see that the phase shift term here is [tex]\( +\frac{8}{3} \)[/tex]. This means the phase shift is effectively [tex]\( -8 \)[/tex] after taking out the [tex]\( \frac{1}{3} \)[/tex].
Thus, the phase shift is:
[tex]\[ -8 \quad \text{to the left} \][/tex]
### Summary:
- The phase shift for [tex]\( y = 3 \sin \left(5\left(x - \frac{\pi}{4}\right)\right) + 2 \)[/tex] is [tex]\( \frac{\pi}{4} \)[/tex] to the right.
- The phase shift for [tex]\( y = -2 \cos \left(\frac{1}{3}(x + 8)\right) + 1 \)[/tex] is [tex]\( -8 \)[/tex] to the left.
