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Answer :
To determine the end behavior of the function [tex]\( y = x^6 - 7x^5 + 7x - 9 \)[/tex], we need to focus on the leading term of the polynomial.
### Step 1: Identify the Leading Term
The leading term of the polynomial function is the term with the highest power of [tex]\( x \)[/tex]. In this case, the leading term is:
[tex]\[ x^6 \][/tex]
### Step 2: Determine the Degree and Coefficient
The function is a polynomial of degree 6, and the leading coefficient is positive (1).
### Step 3: Analyze the End Behavior Based on Degree and Coefficient
The end behavior of a polynomial function is significantly influenced by the degree (even or odd) and the sign of the leading coefficient.
- Even Degree: When a polynomial has an even degree, its end behavior is such that both ends of the graph point in the same direction.
- Positive Leading Coefficient: If the leading coefficient is positive, both ends of the graph will point upwards.
Thus, for the polynomial [tex]\( y = x^6 \)[/tex]:
- As [tex]\( x \to +\infty \)[/tex], [tex]\( y \to +\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to +\infty \)[/tex].
### Conclusion
The end behavior of the function [tex]\( y = x^6 - 7x^5 + 7x - 9 \)[/tex] is:
- As [tex]\( x \to +\infty \)[/tex], [tex]\( y \to +\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to +\infty \)[/tex].
This means that as [tex]\( x \)[/tex] becomes very large in either positive or negative direction, [tex]\( y \)[/tex] also becomes very large and positive.
### Step 1: Identify the Leading Term
The leading term of the polynomial function is the term with the highest power of [tex]\( x \)[/tex]. In this case, the leading term is:
[tex]\[ x^6 \][/tex]
### Step 2: Determine the Degree and Coefficient
The function is a polynomial of degree 6, and the leading coefficient is positive (1).
### Step 3: Analyze the End Behavior Based on Degree and Coefficient
The end behavior of a polynomial function is significantly influenced by the degree (even or odd) and the sign of the leading coefficient.
- Even Degree: When a polynomial has an even degree, its end behavior is such that both ends of the graph point in the same direction.
- Positive Leading Coefficient: If the leading coefficient is positive, both ends of the graph will point upwards.
Thus, for the polynomial [tex]\( y = x^6 \)[/tex]:
- As [tex]\( x \to +\infty \)[/tex], [tex]\( y \to +\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to +\infty \)[/tex].
### Conclusion
The end behavior of the function [tex]\( y = x^6 - 7x^5 + 7x - 9 \)[/tex] is:
- As [tex]\( x \to +\infty \)[/tex], [tex]\( y \to +\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to +\infty \)[/tex].
This means that as [tex]\( x \)[/tex] becomes very large in either positive or negative direction, [tex]\( y \)[/tex] also becomes very large and positive.
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