To find the horizontal shift of the function \( y = 3 \sin(4x - \pi) \), we need to rewrite the argument of the sine function in a way that makes the horizontal shift clear.
The function can be expressed as:
\[ y = 3 \sin(4(x - \frac{\pi}{4})) \]
Here, we can see that the function has been rewritten to reveal the shift. To understand the shift, we look at the inner function \( 4x - \pi \).
To find the value of \( x \) that gives us the horizontal shift, we can set the expression inside the sine function equal to zero:
\[ 4x - \pi = 0 \]
Solving for \( x \):
\[ 4x = \pi \ x = \frac{\pi}{4} \]
Since this is in the form \( f(x - h) \), where \( h \) represents the shift to the right if \( h \) is positive, we find that there is a horizontal shift of \( \frac{\pi}{4} \) units to the right.
So the correct answer is:
start fraction pi over 4 end fraction units to the right