Graphing Rational Functions

Front

What is a rational function?

Back

A rational function is a function that can be expressed as the quotient of two polynomials, typically in the form f(x) = P(x)/Q(x), where P and Q are polynomials.

Front

What are asymptotes in the context of rational functions?

Back

Asymptotes are lines that the graph of a function approaches but never touches. They can be vertical, horizontal, or oblique.

Front

How do you find vertical asymptotes of a rational function?

Back

Vertical asymptotes occur where the denominator of the rational function is equal to zero, provided the numerator is not also zero at that point.

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 determine horizontal asymptotes for rational functions?

Back

To find horizontal asymptotes, compare the degrees of the numerator and denominator: If the degree of the numerator is less, y=0; if equal, y= leading coefficient of P / leading coefficient of Q; if greater, no horizontal asymptote.

Front

What does it mean to shift a function vertically?

Back

Shifting a function vertically means moving it up or down on the graph. For example, f(x) + k shifts the graph up by k units, while f(x) - k shifts it down by k units.

Front

What does it mean to shift a function horizontally?

Back

Shifting a function horizontally means moving it left or right on the graph. For example, f(x - h) shifts the graph right by h units, while f(x + h) shifts it left by h units.

Front

What is a vertical stretch of a function?

Back

A vertical stretch occurs when the output values of a function are multiplied by a factor greater than 1, making the graph taller. For example, f(x) becomes k*f(x) where k > 1.

Front

What is a vertical compression of a function?

Back

A vertical compression occurs when the output values of a function are multiplied by a factor between 0 and 1, making the graph shorter. For example, f(x) becomes k*f(x) where 0 < k < 1.

Front

What is the significance of the denominator in a rational function?

Back

The denominator determines the vertical asymptotes and the domain of the function. It cannot be zero, as this would make the function undefined.