Rational Graphs

Front

What is a rational function?

Back

A rational function is a function that can be expressed as the quotient of two polynomials, where the denominator is not zero.

Front

What is an asymptote?

Back

An asymptote is a line that a graph approaches but never touches or crosses.

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

Front

How do you find the horizontal asymptote of a rational function?

Back

Compare the degrees of the numerator and denominator: If the degree of the numerator is less than the degree of the denominator, y=0; if they are equal, y = leading coefficient of numerator / leading coefficient of denominator; if the numerator's degree is greater, there is no horizontal asymptote.

Front

What is a vertical asymptote?

Back

A vertical asymptote is a vertical line that the graph of a function approaches as the input approaches a certain value, typically where the denominator is zero.

Front

How do you find vertical asymptotes?

Back

Set the denominator of the rational function equal to zero and solve for x.

Front

What is a hole in a graph?

Back

A hole in a graph occurs at a point where a factor in the numerator and denominator cancels out, indicating that the function is undefined at that point.

Front

How do you identify holes in a rational function?

Back

Factor both the numerator and denominator, and find the values of x that make both equal to zero.

Front

What is the significance of the degrees of polynomials in rational functions?

Back

The degrees of the polynomials in the numerator and denominator determine the behavior of the function, including the existence of horizontal and vertical asymptotes.

Front

What does it mean if a rational function has no horizontal asymptote?

Back

It means that as x approaches infinity or negative infinity, the function's value increases or decreases without bound.