The Case for Rational Functions

University of Wisconsin-Eau Claire

https://math.hlasnet.com/research/2016-wmc.html

Rational functions are often difficult for students because they require understanding of zeroes, asymptotes, point discontinuities, and end behaviors. We will investigate behaviors of rational functions symbolically and graphically. Examples will include tasks that indicate what students need to know for work beyond their high school classes.

Rational functions are often difficult for students because they require understanding of zeroes, asymptotes, point discontinuities, and end behaviors. We will investigate behaviors of rational functions symbolically and graphically. Examples will include tasks that indicate what students need to know for work beyond their high school classes.

What do students need beyond high school?

**Rational function:** A quotient of two polynomials

**Polynomial:** A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

\(\dfrac{a}{b} \hspace{2em} \dfrac{x+a}{b} \hspace{2em} \dfrac{a}{x+b} \hspace{2em} \dfrac{x+a}{x+b} \hspace{2em} \ldots \)

**Conjecture:** Sometimes a rational function is zero when numerator is zero

**Reason:** \(\dfrac{P(x)}{Q(x)} = 0\), multiply both sides by \(Q(x)\), so \(P(x) = 0\)

**Exception:** Values must be in domain. Consider \(\dfrac{x^3 + x^2 + x + 1}{x + 1}\). Notice \(x^3 + x^2 + x + 1 = 0\) when \(x = -1\), which is not in the domain. (see point discontinuity)

**Conjecture:** Sometimes a rational function has a vertial asymptote when the denominator is zero

**Reason:** \(\dfrac{P(x)}{Q(x)} = DNE\), when \(Q(x) = 0\)

**Exception:** "Zeros" may factor out. Consider \(\dfrac{(x - 1) (x - 2)}{x - 1} = x - 2\) when \( x \neq 1\). (see point discontinuity)

**Conjecture:** Rational function has a point that is undefined when both the numerator and denomonator are zero.

**Reason:** Consider \(\dfrac{(x - 1) (x - 2)}{x - 1}\). This simplifies to \(x - 2\) when \( x \neq 1\).

**Exception:** multiplicities in denominator

**Conjecture:** Divide the polynomials to identify how the function behaves as \(x \to \infty\)

**Reason:** After dividing the remainder goes to zero for large x values, e.g., \( \dfrac{1+x+x^2}{1+x} = x + \dfrac{1}{1+x} \)

**Conjecture:** There are other ways to write rational functions.

exact answers, bad example: \(x^2 = 2\) when \(x = 1.414\)

algebra skills, \(x = 0\) vs. \(f(x) = 0\)

math questions and healthy skepticism

conjectures, then justification

(DO NOT WANT) memorized rules like horizontal asymptotes

What do students need beyond highschool?

- ability to find patterns

- ability to disect complex ideas

Plesantly surprised

Removeable discontinuities could be better

Works off line