1.7 Rational functions and end behavior

What is a rational function?

How does the degree of the polynomials in the numerator and denominator of a rational function affect its end behavior?

What happens to the end behavior of a rational function when the polynomial in the numerator dominates the polynomial in the denominator for input values of large magnitude?

What happens to the end behavior of a rational function when neither polynomial in the rational function dominates the other for input values of large magnitude?

What is the significance of a horizontal asymptote in a rational function?

Given the rational function f(x) = \dfrac{x^3 - 2x^2 + x - 1}{x^2 - x + 1}, describe the end behavior of the function.

For the function f(x) = \dfrac{3x^2 - 2x + 1}{x^3 - x + 1}, determine which polynomial dominates for input values of large magnitude and describe the end behavior of the function.

Given the rational function f(x) = \dfrac{5x^3 - 3x^2 + 2}{x^3 - 2x^2 + 3x - 1}, determine the end behavior of the function.

For the function f(x) = \dfrac{x^2 - 3x + 2}{4x^2 - x + 1}, which polynomial dominates for input values of large magnitude? Describe the end behavior of the function.

Given the rational function f(x) = \dfrac{2x^2 - 3x + 1}{3x^2 - 2x + 1}, what is the location of its horizontal asymptote?

For the function f(x) = \dfrac{x^3 - x^2 + x - 1}{2x^3 - x^2 + 2}, what is the location of its horizontal asymptote?

Given the rational function f(x) = \dfrac{3x^2 - x + 1}{x^3 - 2x^2 + x - 1}, does it have a slant asymptote? If so, determine its equation.

For the function f(x) = \dfrac{x^3 - 2x^2 + x - 1}{2x^2 - x + 1}, does it have a slant asymptote? If so, determine its equation.

Consider the rational function f(x) = \dfrac{x^3 - 2x^2 + x - 1}{x^2 - x + 1}. How does the end behavior of this function differ from a polynomial function with the same degree? Explain your answer.

Suppose a rational function f(x) has a horizontal asymptote at y = 5 and a slant asymptote. Is this possible? If so, describe what the function might look like. If not, explain why not.

Let's say a new rational function g(x) is created by adding a constant to the numerator of f(x) = \dfrac{x^3 - 2x^2 + x - 1}{x^2 - x + 1}. How would this affect the horizontal asymptote and the end behavior of the function? Explain your reasoning.