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QUESTION

How can you determine the end behavior of a polynomial?

Behavior is dependent on a polynomial's degree.

Before anything else, it's important that we're all on the same page about the concept of a polynomial's degree. A polynomial's degree is the degree of its leading term, that is, the largest exponent of x , in most polynomials.

For example, the polynomial x^2+x+1 is a 2nd degree polynomial since the largest power of x is x^2. Similarly, x^7+25x^4+1000x is a 7th degree polynomial, since the largest power of x is x^7. The only thing that you need consider is the x - all the coefficients are irrelevant to a polynomial's degree.

In general, it's important to notice that polynomials of even degree (i.e. 2nd, 4th, 6th, etc.) all behave similarly, with regards to their end behaviors; polynomials of odd degree (1st, 3rd, 5th, etc.) behave similarly to one another, but differently from even degree polynomials.

Overall, polynomials of even degree all behave similarly to a quadratic polynomial, that is, the parabola.

Above is an image of graphs of x^2 (red), x^4 (green), x^6 (yellow), all even degree polynomials. It's easy to see that they're all very similar graphs, just getting steeper or narrower. The of even degree polynomials is then that they tend to infinity as x approaches either positive or negative infinity.

*A reflection of x about the x-axis, that is, for example -x^2 would merely cause the graph to tend to negative infinity on either end of the graph.

On the other hand, polynomials of odd degree, such as x^1, x^3, x^5, all behave similarly.

Above are the graphs of x^1 (red), x^3 (yellow), and x^5 (green), all odd degree polynomials. We can see that their end behaviors are all the same, in that as x approaches negative infinity, all their function values tend to negative infinity; as x approaches positive infinity, the functions tend to positive infinity as well.

Similar to even degree functions, reflections about the x-axis will merely flip the graphs, changing their end behaviour correspondingly.