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, where the denominator is not zero.

Front

What are vertical asymptotes in rational functions?

Back

Vertical asymptotes are lines x = a where the function approaches infinity or negative infinity as x approaches a. They occur where the denominator of the rational function is zero.

Front

What are horizontal asymptotes in rational functions?

Back

Horizontal asymptotes are lines y = b that the function approaches as x approaches infinity or negative infinity. They indicate the end behavior of the function.

Front

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

Back

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

Front

How do you find the y-intercept of a rational function?

Back

To find the y-intercept, evaluate the function at x = 0, provided the function is defined at that point.

Front

What is the significance of holes in the graph of a rational function?

Back

Holes occur at points where both the numerator and denominator are zero. They indicate removable discontinuities in the function.

Front

What is the end behavior of a rational function?

Back

The end behavior describes how the function behaves as x approaches positive or negative infinity, often determined by the degrees of the numerator and denominator.

Front

How do you determine the degree of a rational function?

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The degree of a rational function is the highest power of x in the numerator or denominator.

Front

What is the relationship between the degrees of the numerator and denominator in determining horizontal asymptotes?

Back

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If they are equal, the asymptote is y = leading coefficient of numerator / leading coefficient of denominator.

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. It can be found using polynomial long division.