Question

sec 5.3 graphs of polynomial functions: problem 10
(1 point)
determine the end behavior for ( y=-7 x^{9}+6 x^{4}+9 x ).

• as ( x

ightarrow-infty, y
ightarrow ) help (numbers)

• as ( x

ightarrow infty, y
ightarrow ) help (numbers)
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Explanation:

Step1: Determine the leading term

The leading term of the polynomial \(y = -7x^{9}+6x^{4}+9x\) is \(-7x^{9}\).

Step2: Analyze the degree and leading coefficient

The degree \(n = 9\) (odd) and the leading coefficient \(a=-7\) (negative).
When \(n\) is odd and \(a<0\):

• As \(x\to-\infty\), \(y = -7x^{9}\to+\infty\) (since \(x^{9}\to-\infty\) when \(x\to-\infty\) and multiplying by \(- 7\) flips the sign).
• As \(x\to+\infty\), \(x^{9}\to+\infty\) and \(y=-7x^{9}\to-\infty\) (because of the negative coefficient).

Answer:

As \(x\to-\infty,y\to+\infty\); As \(x\to+\infty,y\to-\infty\)