Poles and asymptotes of rational functions

What is it about?

A rational function is a quotient of two polynomials. Here it is given in linear factors, for example

$f(x) = \frac{(x-1)(x+1)}{x-2}.$

The zeros of the denominator are not in the domain, here $D = \mathbb{R} \setminus \{2\}$. What happens at such a gap in the domain depends on the numerator:

• Pole: the factor does not cancel (completely), for example $\frac{x-1}{(x-1)^2} = \frac{1}{x-1}$. The graph approaches the vertical asymptote $x = 2$, and the function values tend to $+\infty$ or $-\infty$. If the denominator factor occurs once (in general: an odd number of times), the sign changes (with sign change); for a double factor like $(x-1)^2$ it does not (without sign change).
• Removable gap: numerator and denominator share a linear factor, and it occurs in the numerator at least as often as in the denominator. After cancelling, $f$ can be extended continuously there; the graph keeps a hole. Example: $\frac{(x-1)(x+1)}{x-1} = x + 1$ for $x \neq 1$, hole at $(1 \mid 2)$. Cancelling does not change the domain.

For $x \to \pm\infty$ compare the degree $m$ of the numerator with the degree $n$ of the denominator:

Polynomial division splits $f(x) = s(x) + \frac{r(x)}{q(x)}$ with a remainder $r$ whose degree is smaller than that of the denominator $q$. The fraction $\frac{r(x)}{q(x)}$ tends to $0$ as $x \to \pm\infty$, so the graph approaches $y = s(x)$. Example: $(x^2 - 1) : (x - 2) = x + 2$ remainder $3$, so $f(x) = x + 2 + \frac{3}{x-2}$ and $y = x + 2$ is the oblique asymptote. At $x = 40$ the distance is only $\frac{3}{38} \approx 0.079$.

Try it

1. One pole: drag the green bead (zero of the numerator, a dot) and the red ring (zero of the denominator, a circle with a cross) along the $x$-axis. What happens to the asymptote $y = 1$, and to the arrows at the pole?
2. Gap instead of pole: load this example and drag the bead at $x = 2$ into the ring at $x = 1$. Watch the term: both factors are crossed out, and the vertical asymptote shrinks to a hole. Drag the bead out again.
3. Double pole: why do both branches point upwards? Compare with the sign strip along the $x$-axis.
4. Oblique asymptote: press “Polynomial division step by step” and follow the working as on the board. Then let a point run along the graph: the arrow shows the deviation $f(x) - s(x) = \frac{3}{x-2}$.
5. View from afar: switch “View from afar” on and off. Why does the graph almost merge with the line?
6. Numerator of degree 3: which curve approximates the graph for large $|x|$?
7. Table of values at the pole: how do the values change as $x$ gets closer and closer to $-2$? Compare with the gap: there the values tend to a number.
8. Sign chart: at which points can the sign of $f$ change – and where does it not change? (In the chart, “N” marks numerator factors and “D” denominator factors.)

Tasks

Task 1: Profile

Investigate $f(x) = \frac{x-1}{x+2}$: domain, zero, pole (with or without sign change), asymptotes and $f(0)$. Load task

Task 2: Hole or pole?

Investigate $f(x) = \frac{x^2-1}{x-1}$ at $x = 1$. Load task

Task 3: Polynomial division

Find the asymptote of $f(x) = \frac{x^2-1}{x-2}$. Load task

Task 4: Table of values at the pole

For $f(x) = \frac{x - 1}{x + 2}$ calculate $f(-2.01)$ and $f(-1.99)$. What does this tell you about the behaviour at the pole? Load task

Task 5: Approximating parabola

Show that $f(x) = \frac{0.5\,(x + 2)\,x\,(x - 2)}{x + 1} = 0.5x^2 - 0.5x - 1.5 + \frac{1.5}{x+1}$. Load task

Task 6: May the graph cross the asymptote?

Find out whether the graph of $f(x) = \frac{(x+1)(x-3)}{(x+3)(x-1)}$ crosses its horizontal asymptote. Load task

Task 7: Find a function

Give a rational function with a pole without sign change at $x = 1$ and the horizontal asymptote $y = 2$.

Ideas for the classroom

• Use: in upper secondary (rational functions – limits and asymptotes, later investigating rational functions). Whether approximating curves (numerator degree greater than denominator degree + 1) are required there, please check the curriculum.
• Discovery: students drag a numerator bead into a pole ring and describe what happens to the term and the graph – the idea of a removable gap can grow out of this.
• Profiles: with “Show the properties” off, the profile, poles, gaps, asymptotes, the cancelled term and the table of values stay hidden; the graph remains visible as a check. After working it out, “Reveal the properties” shows everything. Task 7 works well in pairs: one person gives properties, the other builds a matching function.
• Signs without calculation: the sign can only change at zeros and poles, and only for odd multiplicity. The sign strip and the sign chart make this visible.
• Model: numerator and denominator split completely into linear factors with zeros on half units; denominators like $x^2 + 1$ do not occur.
• Common stumbling blocks:
  • Every zero of the denominator is automatically called a pole (cancelling forgotten).
  • The gap is “forgotten” after cancelling – cancelling does not change the domain.
  • For equal degrees, $y = 1$ is given instead of the ratio of the leading coefficients.
  • “A graph may never cross its asymptote.” – It may, see Task 6.
  • In polynomial division, placeholders like $0x$ are missing and the columns slip.