Let's go through the solution for the given function step-by-step.
Function:
[tex]\[
h(x) = -x^4 + 10x^2 - 9
\][/tex]
### Part 1 of 4: End Behavior
To determine the end behavior of the function, we consider the leading term of the polynomial, which is [tex]\(-x^4\)[/tex].
- For [tex]\( x \to \infty \)[/tex]: The term [tex]\(-x^4\)[/tex] will dominate, and since it is negative, [tex]\( h(x) \)[/tex] will approach [tex]\(-\infty \)[/tex].
- For [tex]\( x \to -\infty \)[/tex]: Similarly, [tex]\(-x^4\)[/tex] will again dominate, and [tex]\( h(x) \)[/tex] will approach [tex]\(-\infty \)[/tex].
Thus, the end behavior is:
The end behavior of the function is down to the left and down to the right.
### Part 2 of 4: Finding the [tex]\( y \)[/tex]-Intercept
A [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex].
Substitute [tex]\( x = 0 \)[/tex] into [tex]\( h(x) \)[/tex]:
[tex]\[
h(0) = -0^4 + 10 \cdot 0^2 - 9 = -9
\][/tex]
Thus, the [tex]\( y \)[/tex]-intercept is:
[tex]\[ \boxed{-9} \][/tex]
### Part 3 of 4: Finding the Real Zeros
The real zeros of the function are the values of [tex]\( x \)[/tex] where [tex]\( h(x) = 0 \)[/tex].
Solve:
[tex]\[
-x^4 + 10x^2 - 9 = 0
\][/tex]
The real zeros of this polynomial equation are:
[tex]\[ \boxed{-3, -1, 1, 3} \][/tex]
