Think about what happens to a polynomial such as y = 3x + 2 if you let x equal a very large number, such as 1 000 000 000, or a very small number, such as -20 000 000 000.
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y = 3x + 2 |
By substituting a very large number for x, you see that the y-value will be a very large number also. This means that, as x gets closer to positive infinity, so does y. |
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y = 3x + 2 |
By substituting a very small number for x, you see that the y-value will be a very small number also. This means that, as x gets closer to negative infinity, so does y. |
The description of what happens to the y-value as x approaches positive or negative infinity is called the end behaviour of the function.
Something that is true for all polynomials is that the leading term dominates the behaviour of the polynomial. This means that the leading term gives you all the information you need to determine the end behaviour of a polynomial function. There are four possible end behaviour patterns for a polynomial function. The degree and coefficient of the leading term tell you into which one of the four patterns a polynomial fits.
When you discuss end behaviour, grouping polynomials based on their degree is useful. Polynomials in the same group exhibit the same end behaviour patterns. Quadratics are called even because their degree, 2, is an even number. Linear and cubic are called odd because their degrees, 1 and 3, are odd numbers. These groupings are used in the table below to show the four possible end behaviour patterns.
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