To address the question of whether reflecting the graph of [tex]\( y = \sin x \)[/tex] across the [tex]\( y \)[/tex]-axis is the same as reflecting it across the [tex]\( x \)[/tex]-axis, let's examine the effects of these reflections on the function.
1. Reflecting across the [tex]\( y \)[/tex]-axis:
When we reflect the function [tex]\( y = \sin x \)[/tex] across the [tex]\( y \)[/tex]-axis, we replace [tex]\( x \)[/tex] with [tex]\( -x \)[/tex]. This gives us the new function:
[tex]\[ y = \sin(-x) \][/tex]
2. Reflecting across the [tex]\( x \)[/tex]-axis:
When we reflect the function [tex]\( y = \sin x \)[/tex] across the [tex]\( x \)[/tex]-axis, we replace [tex]\( y \)[/tex] with [tex]\( -y \)[/tex]. This gives us the new function:
[tex]\[ -y = \sin x \][/tex]
which we can rewrite as:
[tex]\[ y = -\sin x \][/tex]
Next, let's compare [tex]\( \sin(-x) \)[/tex] and [tex]\(-\sin(x)\)[/tex].
- The sine function has a property that it is an odd function. This means that:
[tex]\[ \sin(-x) = -\sin(x) \][/tex]
Given this property, we can see that:
[tex]\[ \sin(-x) = -\sin(x) \][/tex]
Therefore, reflecting the graph of [tex]\( y = \sin x \)[/tex] across the [tex]\( y \)[/tex]-axis (which gives [tex]\( y = \sin(-x) \)[/tex]) yields the same result as reflecting it across the [tex]\( x \)[/tex]-axis (which gives [tex]\( y = -\sin x \)[/tex]).
Based on these observations, the statement "Reflecting the graph of [tex]\( y = \sin x \)[/tex] across the [tex]\( y \)[/tex]-axis is the same as reflecting it across the [tex]\( x \)[/tex]-axis" is:
A. True
