Question 1204299: What two parts of a polynomial function indicate the end behaviours of the graph? Found 2 solutions by greenestamps, math_tutor2020:Answer by greenestamps(13200) (Show Source):
The degree of the polynomial, and the leading coefficient.
(1a) Even degree, leading coefficient positive --> upward both left and right
(1b) Even degree, leading coefficient negative --> downward both left and right
(2a) Odd degree, leading coefficient positive --> downward to the left, upward to the right
(2b) Odd degree, leading coefficient negative --> upward to the left, downward to the right
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The two things we look for are: the leading coefficient and the degree of the polynomial.
Both of those help form the leading term.
The other terms won't affect the end behavior.
Something like y = 6x^3 + 10x^2 + 5x + 7 has the same end behavior as y = 6x^3.
This is because as x gets really really large, the leading term has the most effect compared to the other terms.
Cubing a large number makes it much bigger compared to squaring it for example.
Consider the monomial y = a*x^n
If a > 0 and n is even, then the two endpoints rise up together.
Example: y = 3x^2
If a < 0 and n is even, then the endpoints fall together
Example: y = -3x^2
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If n is odd, then the endpoints point in opposite directions.
a > 0 has the end behavior "fall to the left, rise to the right"
Example: y = 2x^5
a < 0 has the end behavior "rise to the left, fall to the right" when n is odd.
Example: y = -5x^3
I encourage you to try out other examples. Use Desmos or GeoGebra as a graphing tool.
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