To determine the [tex]\( y \)[/tex]-intercept of the graph of the function [tex]\( y = -x^6 - 6x^5 + 50x^3 + 45x^2 - 108x - 108 \)[/tex], we need to find the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 0.
The [tex]\( y \)[/tex]-intercept is the point where the graph intersects the [tex]\( y \)[/tex]-axis, which happens when [tex]\( x = 0 \)[/tex].
Let's substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[
y = -0^6 - 6 \cdot 0^5 + 50 \cdot 0^3 + 45 \cdot 0^2 - 108 \cdot 0 - 108
\][/tex]
Calculating each term:
[tex]\[
0^6 = 0
\][/tex]
[tex]\[
6 \cdot 0^5 = 0
\][/tex]
[tex]\[
50 \cdot 0^3 = 0
\][/tex]
[tex]\[
45 \cdot 0^2 = 0
\][/tex]
[tex]\[
108 \cdot 0 = 0
\][/tex]
Thus, the equation simplifies to:
[tex]\[
y = 0 - 0 + 0 + 0 - 0 - 108
\][/tex]
This simplifies further to:
[tex]\[
y = -108
\][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept of the graph is [tex]\( (0, -108) \)[/tex].
So, the correct answer is:
A. [tex]\((0, -108)\)[/tex]
