Graphs of Rational Functions

Front

What is a vertical asymptote?

Back

A vertical asymptote is a line x = a where a rational function approaches infinity or negative infinity as the input approaches a. It indicates values that the function cannot take.

Front

How do you find vertical asymptotes in a rational function?

Back

Vertical asymptotes can be found by setting the denominator of the rational function equal to zero and solving for x.

Front

What is a hole in the graph of a rational function?

Back

A hole occurs at a point where a factor in the numerator and denominator cancels out, indicating that the function is not defined at that point.

Front

How do you identify holes in a rational function?

Back

To identify holes, factor both the numerator and denominator, and find common factors that cancel out.

Front

What are x-intercepts in a graph?

Back

X-intercepts are points where the graph crosses the x-axis, which occur when the output of the function is zero.

Front

How do you find x-intercepts of a rational function?

Back

To find x-intercepts, set the numerator of the rational function equal to zero and solve for x.

Front

What is the significance of the degree of the numerator and denominator in rational functions?

Back

The degree of the numerator and denominator helps determine the end behavior of the graph and the presence of horizontal asymptotes.

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a line y = b that the graph of a rational function approaches as x approaches infinity or negative infinity.

Front

How do you find horizontal asymptotes in a rational function?

Back

Horizontal asymptotes can be found by comparing the degrees of the numerator and denominator: if the degree of the numerator is less, y=0; if equal, y=b/a; if greater, no horizontal asymptote.

Front

What is the relationship between holes and vertical asymptotes?

Back

Holes occur at points where a factor cancels out, while vertical asymptotes occur at points where the denominator is zero but the factor does not cancel.