Answer:
- +x^(any) → ∞ for x → ∞
- -x^(any) → -∞ for x → ∞
- x^(even) → (-x)^(even) for x → -∞
- x^(odd) → -(-x)^(odd) for x → -∞
Step-by-step explanation:
You want a description of the end behavior of even- and odd-degree polynomials with positive and negative leading coefficients.
Infinity
As x gets large (approaches infinity), any power of x will also get large (approach infinity). The sign of the infinity being approached for large positive x will match the sign of the leading coefficient.
Even degree
When the degree of the polynomial is even, the right-end and left-end behaviors match.
Odd degree
When the degree of the polynomial is odd, the sign of the left-end behavior is opposite that of the right end behavior.
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Additional comment
You can think of any even power of x as matching the end-behavior of |x|. Similarly, any odd power of x will match the end behavior of x. The general trend of even-degree polynomials with a positive leading coefficient is a U- or V-shape. The general trend of any odd-degree polynomial with a positive leading coefficient is a /-shape (rising, left-to-right). A negative leading coefficient turns these shapes upside down.
When it comes to end behavior, the leading term is the only one that needs to be considered.
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