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, 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, the asymptote is y=0; if equal, y= leading coefficient of numerator/leading coefficient of denominator; if 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 function's value approaches infinity or negative infinity.

Front

How do we find vertical asymptotes in a rational function?

Back

Vertical asymptotes occur at the values of x that make the denominator zero, provided these zeros are non-removable.

Front

What is a slant (oblique) asymptote?

Back

A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator.

Front

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

Back

X-intercepts are found by setting the numerator equal to zero and solving for x.

Front

What are the x-intercepts of the function f(x) = (x-4)(x+2)/(x^2+1)?

Back

The x-intercepts are x=4 and x=-2.

Front

What is a removable discontinuity?

Back

A removable discontinuity occurs when a factor in the numerator and denominator cancels out, leading to a hole in the graph.