Graphing and Transformations of Rational Functions Refresher

Front

What is a vertical asymptote?

Back

A vertical asymptote is a line x = a where a function approaches infinity or negative infinity as x approaches a. It occurs when the denominator of a rational function equals zero and the numerator does not.

Front

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

Back

To find the vertical asymptote of a rational function, set the denominator equal to zero and solve for x.

Front

What does |a| > 1 indicate about the graph of f(x) = \frac{a}{x-h} + k?

Back

It indicates a vertical stretch of the graph.

Front

What is a hole in a rational function?

Back

A hole occurs at a point where both the numerator and denominator of a rational function equal zero, indicating that the function is undefined at that point.

Front

How do you identify a hole in a rational function?

Back

To identify a hole, factor both the numerator and denominator, and find the common factors. The x-value that makes the common factor zero is where the hole occurs.

Front

What transformation does -f(x) represent?

Back

The transformation -f(x) represents a reflection of the graph of f(x) across the x-axis.

Front

What is the graph of f(x) = \frac{1}{x} known as?

Back

The graph of f(x) = \frac{1}{x} is known as a hyperbola.

Front

What is the horizontal asymptote of a rational function?

Back

The horizontal asymptote is a horizontal line that the graph approaches as x approaches infinity or negative infinity. It is determined by the degrees of the numerator and denominator.

Front

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

Back

1. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. 2. If the degrees are equal, the horizontal asymptote is y = \frac{leading coefficient of numerator}{leading coefficient of denominator}. 3. If the degree of the numerator is greater, there is no horizontal asymptote.

Front

What does a positive leading coefficient indicate about the end behavior of a rational function?

Back

A positive leading coefficient indicates that as x approaches positive infinity, the function approaches positive infinity, and as x approaches negative infinity, the function approaches negative infinity.