To answer this question, we need to understand how the transformation from [tex]\( y = f(x) \)[/tex] to [tex]\( y = -f(x) \)[/tex] affects the coordinates of the points on the graph. Specifically, we need to determine what happens to the point [tex]\((7, 0)\)[/tex].
The original point given is [tex]\((7,0)\)[/tex], meaning that when [tex]\( x = 7 \)[/tex], [tex]\( y = f(7) = 0 \)[/tex].
When we transform the function to [tex]\( y = -f(x) \)[/tex], the [tex]\( y \)[/tex]-coordinate of each point on the graph of [tex]\( y = f(x) \)[/tex] is negated.
To find the corresponding point on the graph of [tex]\( y = -f(x) \)[/tex] for the given point [tex]\((7,0)\)[/tex]:
1. Identify the [tex]\( x \)[/tex]-coordinate. It remains unchanged, so it is still [tex]\( 7 \)[/tex].
2. Negate the [tex]\( y \)[/tex]-coordinate. The original [tex]\( y \)[/tex]-coordinate is [tex]\( 0 \)[/tex], so negating it gives us [tex]\( -0 \)[/tex].
Mathematically, [tex]\( -0 \)[/tex] is simply [tex]\( 0 \)[/tex].
So, the point [tex]\((7, 0)\)[/tex] on the graph of [tex]\( y = f(x) \)[/tex] corresponds to the point [tex]\((7, 0)\)[/tex] on the graph of [tex]\( y = -f(x) \)[/tex].
Therefore, the correct answer is:
[tex]\[
(7,0)
\][/tex]
