4.4 — Are You Rational?

1. In class: Definitions and a guide to the topics

This section brings together several demanding topics. Start with the definitions below, then use the videos organized by topic for explanations and worked examples. Pause and try examples yourself before watching the solutions.

Rational numbers, expressions, and functions — Main 1–3

A rational number is a number that can be expressed as the quotient or fraction \(\frac pq\) of two integers, a numerator \(p\) and a nonzero denominator \(q\). For example, \(\frac23\), \(-5=\frac{-5}{1}\), and \(2.7=\frac{27}{10}\) are rational numbers.

A polynomial is a sum of terms such as \(3x^2\), \(-2x\), and 5, using only nonnegative whole-number powers of the variable. A rational expression is a fraction whose numerator and denominator are polynomials. A rational function is a function that can be written as \(f(x)=\frac{P(x)}{Q(x)}\), where \(P\) and \(Q\) are polynomials and \(Q\) is not the zero polynomial. Exclude any input that makes the denominator zero.

Examples include \(\frac1x\), \(\frac{x+1}{x-2}\), and \(\frac{x^2+1}{x^2+4}\). “Rational function” describes the form of the rule; it does not mean every output must be a rational number.

Simplifying and keeping excluded inputs — Main 4–7

A factor is multiplied by another expression; a term is part of a sum. Simplifying a rational expression means factoring and canceling common nonzero factors, not crossing out terms in a sum. Keep every excluded input from the original denominator, even after a factor cancels.

Study factoring and cancellation in the simplifying videos below.

Holes versus vertical asymptotes — Main 8–9 and Go

A hole is a missing point on the graph at a finite height. It occurs when an excluded input has a finite output in the simplified rule. Find its height by substituting that input into the simplified rule, and mark it with an open circle.

A vertical asymptote is a line \(x=a\) that the curve approaches while rising or falling without bound. After canceling common factors, real zeros of the remaining denominator locate the vertical asymptotes. An excluded input is not automatically a vertical asymptote: it may be a hole instead. See the holes and asymptotes video.

Proper and improper rational expressions — Main 10–11 and Set

The degree of a polynomial is its highest power with a nonzero coefficient. A rational expression is proper when the numerator’s degree is smaller than the denominator’s degree. It is improper when the numerator’s degree is equal to or greater than the denominator’s degree. Compare degrees, not the values at a particular input.

For example, \(\frac{x+1}{x^2+1}\) is proper; \(\frac{x^2+1}{x+1}\) is improper. A proper rational function approaches 0 at both ends, so its horizontal asymptote is \(y=0\).

Quotient and remainder — Main 12–16 and Set

Polynomial long division rewrites an improper expression as a quotient plus a remainder fraction:

Here \(q(x)\) is the quotient and \(r(x)\) is the remainder, whose degree is smaller than the denominator’s degree (or the remainder is zero). Keep the original denominator exclusions. Review the long-division video before attempting the packet’s divisions.

Slant asymptotes — Main 16–19

A slant, or oblique, asymptote is a sloping line that the graph gets closer and closer to as you move far to the left or right. This is the new asymptote topic in 4.4; vertical and horizontal asymptotes appeared earlier.

When the numerator’s degree is exactly one greater than the denominator’s degree, division gives a linear quotient. That line is the slant asymptote because the remainder fraction approaches 0. For example:

As \(x\) becomes very large in either direction, \(1/(x+2)\) approaches 0, so the graph gets closer to \(y=x+3\). A larger degree difference can give a higher-degree polynomial quotient, rather than a slant line. Watch the horizontal and slant asymptotes video for a worked explanation.

2. Ready — Compare a polynomial with its reciprocal

Review 4.2: zeros and domains. Polynomial functions allow every real input. Their reciprocals exclude real polynomial zeros and never have output zero. Try Ready before checking.

3. Set — Classify and divide rational expressions

Use the degree definitions and division procedure. A polynomial alone is a rational function with denominator 1, of degree 0. In 15–18 the packet specifically asks for long division when the expression is improper.

4. Go — Retain exclusions and mark holes

Use the original-domain rule and hole procedure. Try the domains, asymptotes and graph annotations before checking. Every canceled denominator zero still stays outside the original domain.

Teacher/parent review of the source packet

Main numbering, pages 24–25: 16 is duplicated. The two prompts are labeled by page throughout this lesson.

Corrections — Page 25 introduction and Main 19 teacher guidance

At the top of page 25, the student packet says: “we did not consider the case when the numerator of the fraction is greater than the denominator.” Here it should say the degree of the numerator is greater than the degree of the denominator. Degree means the highest power of the variable; it is not a comparison of the numerical values of the two expressions at a particular input.

Question 19 then asks students to use the two preceding examples to describe how to graph rational functions when the numerator’s degree is greater. Both examples have a degree-2 numerator and a degree-1 denominator. After division, each has a sloping line plus a fraction that approaches 0, so each graph has a slant asymptote.

The teacher’s guide’s “Discuss—Part 2” instructions for Question 19 include “Using the equivalent expression to identify the slant asymptote.” That instruction needs a qualification: a greater numerator degree does not always give a slant line. The degree difference must be exactly 1.

For example, \(\frac{x^2+1}{x}=x+\frac1x\) gets closer to the line \(y=x\), so it has a slant asymptote. But \(\frac{x^3+1}{x}=x^2+\frac1x\) gets closer to the parabola \(y=x^2\), not a line. In both examples the numerator’s degree is greater; only the first has a slant asymptote.

More precise wording: Divide first and examine the quotient. If it is linear, its graph is the slant asymptote. If it has degree 2 or higher, the ends of the function approach that polynomial curve instead.

Go b, page 28: “open circle anywhere undefined” needs a finite-point qualification. Mark canceled-factor holes with open circles and remaining vertical asymptotes with dashed lines. Main 4/5 and Go must retain every original denominator exclusion after cancellation.

Coverage