To find the [tex]\( y \)[/tex]-intercept of the graph of the function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex], we need to determine the point at which the graph intersects the [tex]\( y \)[/tex]-axis. The [tex]\( y \)[/tex]-intercept occurs when [tex]\( x = 0 \)[/tex].
Let's substitute [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 0^2 + 3(0) + 5 \][/tex]
Simplify the right-hand side:
[tex]\[ f(0) = 0 + 0 + 5 \][/tex]
[tex]\[ f(0) = 5 \][/tex]
So, the [tex]\( y \)[/tex]-coordinate of the [tex]\( y \)[/tex]-intercept is 5. Therefore, the [tex]\( y \)[/tex]-intercept is the point [tex]\((0, 5)\)[/tex].
Thus, the answer is:
[tex]\((0, 5)\)[/tex]
