Description

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.

Description

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?

Definitions

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.

What is simpliest rational function?

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

Summary: Zeros

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)

Summary: Vertical asympototes

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)

Removable ("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

End behaviors

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.

Desmos example

What I want as a professor:

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

Last thing: Desmos review

Plesantly surprised

Removeable discontinuities could be better

Works off line