To find the coordinates of the [tex]\( y \)[/tex]-intercept for the function [tex]\( f(x) = -2x^3 + 13x^2 - 22x + 8 \)[/tex], we need to determine the value of the function when [tex]\( x = 0 \)[/tex].
1. Start by substituting [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(0) = -2(0)^3 + 13(0)^2 - 22(0) + 8
\][/tex]
2. Simplify each term:
- [tex]\( -2(0)^3 = 0 \)[/tex]
- [tex]\( 13(0)^2 = 0 \)[/tex]
- [tex]\( -22(0) = 0 \)[/tex]
- [tex]\( 8 \)[/tex] remains as it is
3. Summing these simplified terms together:
[tex]\[
f(0) = 0 + 0 + 0 + 8 = 8
\][/tex]
Therefore, the function [tex]\( f(x) \)[/tex] evaluates to 8 when [tex]\( x = 0 \)[/tex].
The [tex]\( y \)[/tex]-intercept is the point where the graph of the function crosses the [tex]\( y \)[/tex]-axis. This occurs when [tex]\( x = 0 \)[/tex]. From the calculation, we've determined that when [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 8 \)[/tex]. Hence, the coordinates of the [tex]\( y \)[/tex]-intercept are:
[tex]\[
(0, 8)
\][/tex]
So, the coordinates of the [tex]\( y \)[/tex]-intercept are [tex]\( (0, 8) \)[/tex].
