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Answer :
To determine the end behavior of the polynomial function [tex]\( y = 9x^4 + 9x^2 \)[/tex], we should focus on the leading term of the polynomial, which is the term with the highest power of [tex]\( x \)[/tex].
1. Identify the Leading Term:
- The given polynomial is [tex]\( y = 9x^4 + 9x^2 \)[/tex].
- The leading term is [tex]\( 9x^4 \)[/tex], as it has the highest exponent.
2. Analyze the Leading Term:
- The exponent of the leading term is 4, which is an even number.
- The coefficient of the leading term is positive (9).
3. End Behavior of Polynomials:
- For polynomials, the end behavior depends on the degree (exponent) and the leading coefficient of the term with the highest power.
- If the degree is even and the leading coefficient is positive, the ends of the graph of the polynomial will rise as [tex]\( x \)[/tex] approaches both positive and negative infinity.
4. Conclusion for the Given Polynomial:
- As [tex]\( x \to \infty \)[/tex] (x goes to positive infinity), [tex]\( y \to \infty \)[/tex]. This is because the leading term's positive coefficient and even exponent lead the graph upward.
- As [tex]\( x \to -\infty \)[/tex] (x goes to negative infinity), [tex]\( y \to \infty \)[/tex]. Similarly, an even exponent means that the direction of the graph at both extreme ends is the same; it rises because the leading coefficient is positive.
Therefore, the end behavior of the function [tex]\( y = 9x^4 + 9x^2 \)[/tex] is:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to \infty \)[/tex].
1. Identify the Leading Term:
- The given polynomial is [tex]\( y = 9x^4 + 9x^2 \)[/tex].
- The leading term is [tex]\( 9x^4 \)[/tex], as it has the highest exponent.
2. Analyze the Leading Term:
- The exponent of the leading term is 4, which is an even number.
- The coefficient of the leading term is positive (9).
3. End Behavior of Polynomials:
- For polynomials, the end behavior depends on the degree (exponent) and the leading coefficient of the term with the highest power.
- If the degree is even and the leading coefficient is positive, the ends of the graph of the polynomial will rise as [tex]\( x \)[/tex] approaches both positive and negative infinity.
4. Conclusion for the Given Polynomial:
- As [tex]\( x \to \infty \)[/tex] (x goes to positive infinity), [tex]\( y \to \infty \)[/tex]. This is because the leading term's positive coefficient and even exponent lead the graph upward.
- As [tex]\( x \to -\infty \)[/tex] (x goes to negative infinity), [tex]\( y \to \infty \)[/tex]. Similarly, an even exponent means that the direction of the graph at both extreme ends is the same; it rises because the leading coefficient is positive.
Therefore, the end behavior of the function [tex]\( y = 9x^4 + 9x^2 \)[/tex] is:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to \infty \)[/tex].
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