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Answer :
To determine the end behavior of the polynomial function [tex]\( h(x) = -5x^4 + 7x^3 - 6x^2 + 9x + 2 \)[/tex], we need to analyze its leading term, as this will guide us in understanding how the function behaves as [tex]\( x \)[/tex] approaches positive or negative infinity.
1. Identify the Leading Term:
The leading term in a polynomial is the term with the highest power of [tex]\( x \)[/tex]. In this case, the leading term is [tex]\(-5x^4\)[/tex].
2. Consider the Degree of the Polynomial:
The degree of the polynomial is 4, which is an even number. The degree affects the shape of the graph at the ends (as [tex]\( x \)[/tex] approaches [tex]\(\pm \infty\)[/tex]).
3. Consider the Leading Coefficient:
The leading coefficient is -5, which is negative.
4. Determine the End Behavior:
- As [tex]\( x \to +\infty \)[/tex]: Since the degree is even and the leading coefficient is negative, the function will approach negative infinity. This means that as [tex]\( x \)[/tex] gets larger and larger, [tex]\( h(x) \)[/tex] will decrease without bound.
- As [tex]\( x \to -\infty \)[/tex]: Similarly, because the degree is even and the leading coefficient is negative, the function will also approach negative infinity. This implies that as [tex]\( x \)[/tex] becomes more negative, [tex]\( h(x) \)[/tex] will again decrease without bound.
Therefore, we summarize the end behavior of [tex]\( h(x) \)[/tex] as follows:
- As [tex]\( x \to +\infty \)[/tex], [tex]\( h(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( h(x) \to -\infty \)[/tex].
This means in both directions, the graph of the function falls towards negative infinity.
1. Identify the Leading Term:
The leading term in a polynomial is the term with the highest power of [tex]\( x \)[/tex]. In this case, the leading term is [tex]\(-5x^4\)[/tex].
2. Consider the Degree of the Polynomial:
The degree of the polynomial is 4, which is an even number. The degree affects the shape of the graph at the ends (as [tex]\( x \)[/tex] approaches [tex]\(\pm \infty\)[/tex]).
3. Consider the Leading Coefficient:
The leading coefficient is -5, which is negative.
4. Determine the End Behavior:
- As [tex]\( x \to +\infty \)[/tex]: Since the degree is even and the leading coefficient is negative, the function will approach negative infinity. This means that as [tex]\( x \)[/tex] gets larger and larger, [tex]\( h(x) \)[/tex] will decrease without bound.
- As [tex]\( x \to -\infty \)[/tex]: Similarly, because the degree is even and the leading coefficient is negative, the function will also approach negative infinity. This implies that as [tex]\( x \)[/tex] becomes more negative, [tex]\( h(x) \)[/tex] will again decrease without bound.
Therefore, we summarize the end behavior of [tex]\( h(x) \)[/tex] as follows:
- As [tex]\( x \to +\infty \)[/tex], [tex]\( h(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( h(x) \to -\infty \)[/tex].
This means in both directions, the graph of the function falls towards negative infinity.
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