Question

What is the horizontal shift of the function y equals 3 sine left-parenthesis 4 x minus pi right-parenthesis?
(1 point)
Responses

π units to the right
π units to the right

start fraction pi over 2 end fraction units to the right
Image with alt text: start fraction pi over 2 end fraction units to the right

start fraction pi over 3 end fraction units to the right
Image with alt text: start fraction pi over 3 end fraction units to the right

start fraction pi over 4 end fraction units to the right

Answer 1

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**