Question

What is the end behavior of a function that is a fraction whose polynomial numerator has a power degree than its polynomial denominator

Answer 1

The end behavior of a function where the polynomial numerator has a lower degree than the polynomial denominator is that the function's value will approach zero as the variable x tends to positive or negative infinity.

Step-by-step explanation:

The end behavior of a function that is expressed as a fraction (or rational function) in which the polynomial numerator has a lower degree than the polynomial denominator will approach zero as the input (x) either goes to positive infinity or negative infinity. This is because as the absolute value of x increases, the values in the denominator increase much faster than those in the numerator, causing the entire fraction’s value to decrease towards zero. This concept is central to understanding how the degree of a polynomial affects the end behavior of a rational function.

An example to illustrate this could involve the rational function 1/xn, where n is a positive integer. As x becomes larger and larger, the value of xn grows much faster since it has a higher exponential degree, making the fraction much smaller, thus tending towards zero.

Answer 2

End behavior of a polynomial function is based on the degree of the function and the sign of the leading coefficient.

Sign of the Leading Coefficient determines behavior of right side:

• Positive: right side goes to positive infinity
• Negative: right side goes to negative infinity

Degree of the function determines the behavior of the left side:

• Odd degree: left side is opposite direction of right side
• Even degree: left side is same direction as right side

If you have an expression in the denominator, then you must divide the denominator into the numerator. The result will have a degree and a leading coefficient. Use the rules stated above to determine the end behavior.

For example:

y =
`(x^(2)+2x-3)/(x-1)`

We can factor to get: y =
`((x-1)(x+3))/((x-1))`

y = x + 3

Leading Coefficient of y = x + 3 is positive so right side goes to positive infinity.

Degree of y = x + 3 is odd so left side is opposite direction of right side, which means left side goes to negative infinity.

The denominator may not divide evenly into the numerator thus leaving a remainder, but that is ok. We can still use the rules stated above.