Glossary

Degree (of polynomials)

Criticality: 3

The highest exponent of the variable in a polynomial. Comparing the degrees of the numerator and denominator polynomials is essential for determining a rational function's end behavior.

Example:

In the polynomial $5x^3 - 7x + 1$ , the degree is 3, which is the highest power of x.

End Behavior

Criticality: 3

Describes what happens to the output values of a function as the input (x) approaches positive or negative infinity. For rational functions, it is primarily determined by the comparison of the degrees of the numerator and denominator.

Example:

The end behavior of $f(x) = \frac{2x}{x^2+1}$ is that the function approaches 0 as x gets very large or very small.

Horizontal Asymptote

Criticality: 3

A horizontal line that a rational function approaches as x tends towards positive or negative infinity. Its existence and value depend on the comparison of the degrees of the numerator and denominator polynomials.

Example:

The function $k(x) = \frac{6x^2 - x + 1}{2x^2 + 5}$ has a horizontal asymptote at $y = 3$ , determined by the ratio of the leading coefficients.

Leading Terms

Criticality: 2

The term in a polynomial that contains the highest power of the variable. The ratio of the leading terms of the numerator and denominator polynomials is key to understanding a rational function's end behavior.

Example:

For the rational function $g(x) = \frac{4x^5 - 2x^2}{7x^5 + 9}$ , the leading terms are $4x^5$ and $7x^5$ .

Limits

Criticality: 2

A fundamental concept in calculus used to describe the value that a function or sequence 'approaches' as the input or index approaches some value. It provides a formal way to express end behavior.

Example:

We use limits to write $\lim_{x \to \infty} \frac{1}{x^2} = 0$ , indicating that as x grows infinitely large, the function's value gets arbitrarily close to zero.

Ratio (or Quotient)

Criticality: 1

The result of dividing one quantity by another. In the context of rational functions, it refers to the division of the numerator polynomial by the denominator polynomial.

Example:

When analyzing $f(x) = \frac{3x+2}{x^2-5}$ , we are looking at the ratio of the linear polynomial to the quadratic polynomial.

Rational Function

Criticality: 3

A function that can be expressed as the ratio of two polynomial functions, where the denominator polynomial is not equal to zero.

Example:

The function $f(x) = \frac{x^2 - 4}{x + 1}$ is a rational function because it's a polynomial divided by another polynomial.

Slant Asymptote

Criticality: 3

A diagonal line that a rational function approaches as x tends towards positive or negative infinity. This occurs when the degree of the numerator is exactly one greater than the degree of the denominator.

Example:

The function $h(x) = \frac{x^2 + 2x + 3}{x + 1}$ has a slant asymptote at $y = x + 1$ , which can be found by polynomial long division.

Vertical Asymptotes

Criticality: 2

Vertical lines on a graph where the function's value approaches positive or negative infinity. They occur at x-values where the denominator of a simplified rational function is zero.

Example:

The function $f(x) = \frac{x}{x - 4}$ has a vertical asymptote at $x = 4$ , because the denominator becomes zero at this point.