Problem 79 - Algebra and Trigonometry

Question

What is meant by the end behavior of a polynomial function?

Step-by-Step Solution

Answer

The end behavior of a polynomial function refers to the value that the function approaches as x moves towards positive or negative infinity. It is determined by the degree and leading coefficient of the function. If the degree is even, both ends of the graph will approach the same direction (both going up if the leading coefficient is positive, both going down if negative). If the degree is odd, one end will go up and the other will go down, dependent on the sign of the leading coefficient.

Step 1: Understanding Degree and Leading Coefficient

For any polynomial function represented as \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \( a_n, a_{n-1}, ..., a_0 \) are constants, the coefficient \(a_n\) of the highest power of 'x' is called the 'leading coefficient'. The highest power 'n' is referred to as the 'degree'.

Step 2: Rules of End behavior

For a polynomial function, if the 'degree' is even, as 'x' approaches infinity or negative infinity, 'f(x)' will approach positive infinity if the 'leading coefficient' is positive. If the leading coefficient is negative, 'f(x)' will approach negative infinity. If the 'degree' is odd, as 'x' approaches infinity or negative infinity, 'f(x)' will approach different directions based on the sign of the leading coefficient.

Step 3: Summary

In summary, the end behavior of a polynomial function refers to the value that the function approaches as 'x' moves towards positive or negative infinity, and this is determined by the degree and leading coefficient of the polynomial function.