To analyze the end behavior of the function [tex]\( h(x) = -x^4 + 10x^2 - 9 \)[/tex], we can consider the properties of polynomials with even degrees and leading coefficients.
1. Identifying the Degree and Leading Coefficient:
- The degree of the polynomial [tex]\( h(x) \)[/tex] is 4, which is an even degree.
- The leading term is [tex]\( -x^4 \)[/tex], which has a leading coefficient of [tex]\(-1\)[/tex], and it is negative.
2. Analyzing the End Behavior for Polynomials:
- For polynomials of even degree with a negative leading coefficient, the end behavior will approach negative infinity ([tex]\(-\infty\)[/tex]) as [tex]\( x \)[/tex] moves towards both positive infinity ([tex]\( \infty \)[/tex]) and negative infinity ([tex]\( -\infty \)[/tex]).
- This means:
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex] (the left end), [tex]\( h(x) \)[/tex] will approach [tex]\( -\infty \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] (the right end), [tex]\( h(x) \)[/tex] will also approach [tex]\( -\infty \)[/tex].
So, the end behavior of the function [tex]\( h(x) = -x^4 + 10x^2 - 9 \)[/tex] is:
- To the left (as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]), [tex]\( h(x) \)[/tex] approaches [tex]\( -\infty \)[/tex].
- To the right (as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex]), [tex]\( h(x) \)[/tex] also approaches [tex]\( -\infty \)[/tex].
Therefore, the end behavior of the function is:
- To the left: [tex]\( -\infty \)[/tex]
- To the right: [tex]\( -\infty \)[/tex]
