Let's address the problem step-by-step.
### Part 1 of 4
The End Behavior of the Function
To determine the end behavior, we look at the term with the highest power of [tex]\(x\)[/tex] in the given function [tex]\(h(x) = -x^4 + 10x^2 - 9\)[/tex].
The highest power term is [tex]\(-x^4\)[/tex]. Because the coefficient of [tex]\(x^4\)[/tex] (which is -1) is negative, as [tex]\(x\)[/tex] approaches both [tex]\(+\infty\)[/tex] and [tex]\(-\infty\)[/tex], the term [tex]\(-x^4\)[/tex] will dominate and will go to [tex]\(-\infty\)[/tex]. This means:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( h(x) \to -\infty \)[/tex] (down to the left)
- As [tex]\( x \to +\infty \)[/tex], [tex]\( h(x) \to -\infty \)[/tex] (down to the right)
So, the end behavior of the function is:
down to the left and down to the right.
### Part 2 of 4
Finding the y-Intercept
To find the [tex]\(y\)[/tex]-intercept of a function, we evaluate the function at [tex]\(x = 0\)[/tex].
Given:
[tex]\[ h(x) = -x^4 + 10x^2 - 9 \][/tex]
Substitute [tex]\(x = 0\)[/tex]:
[tex]\[ h(0) = -0^4 + 10(0)^2 - 9 = -9 \][/tex]
So, the [tex]\(y\)[/tex]-intercept is:
[tex]\[ -9 \][/tex]
### Summary of Both Parts
Part 1:
The end behavior of the function is [tex]\(\boxed{\text{down}}\)[/tex] down to the left and [tex]\(\boxed{\text{down}}\)[/tex] down to the right.
Part 2:
The [tex]\(y\)[/tex]-intercept(s) is [tex]\(\boxed{-9}\)[/tex].
Let's proceed to the next part of the problem once we have confirmed these parts.
