Rational Functions-Vertical Asymptotes and Holes

Front

What is a rational function?

Back

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

Front

What is a vertical asymptote?

Back

A vertical asymptote is a vertical line x = a where the function approaches infinity or negative infinity as x approaches a.

Front

How do you find vertical asymptotes in a rational function?

Back

Vertical asymptotes are found by setting the denominator Q(x) = 0 and solving for x.

Front

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

Back

A hole occurs in the graph of a rational function at a point where a factor in the numerator cancels with a factor in the denominator.

Front

How do you identify holes in a rational function?

Back

To identify holes, factor both the numerator and denominator, then find values of x that make both equal to zero.

Front

What is the significance of the degree of the numerator and denominator?

Back

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

Front

What happens to the function at a vertical asymptote?

Back

As the function approaches a vertical asymptote, the function values increase or decrease without bound, leading to infinity or negative infinity.

Front

Can a rational function have both vertical asymptotes and holes?

Back

Yes, a rational function can have both vertical asymptotes and holes, as they arise from different factors in the numerator and denominator.

Front

What is the general form of a vertical asymptote equation?

Back

The equation of a vertical asymptote is written in the form x = a.

Front

What is the relationship between holes and vertical asymptotes?

Back

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