Rational Functions

Rational Function

A rational function keeps a variable in its denominator and begins with inverse proportion y = k/x, where two quantities always multiply to a constant. Its graph is a hyperbola centered at the origin that creeps closer and closer to the axes yet never touches them. Shifting it to y = k/(x − a) + b moves the center to where the asymptotes cross, (a, b). Vary k and the shifts a and b here to watch the hyperbola and its asymptotes move.

Inverse Proportion — Birth of Rational Functions

💧 Faucet & Tub

①Imagine filling a 12-liter tub

②1 L/min → 12 min, 2 L/min → 6 min, 3 L/min → 4 min

③As flow x grows, time y shrinks: xy = 12 (constant!)

④This is inverse proportion — the graph y = k/x

k (constant)2

Basic Rational Function

y = kx (x ≠ 0)

xy = k: product of variables stays constant

📐 Properties of the Basic Form

①Origin symmetry: f(−x) = −f(x) (odd function)

②k > 0 → quadrants 1 & 3; k < 0 → quadrants 2 & 4

③Larger |k| pushes the graph farther from the axes

④Asymptotes: y = 0 (x-axis) and x = 0 (y-axis) — never touched

Translation — Asymptote Intersection is Key

a (vertical asymptote)1

b (horizontal asymptote)1

Standard Form

y = kx − a + b

Vertical asymptote x = a, horizontal y = b

🔑 Why the Intersection Matters

①y = k/x has 'center' (0, 0)

②y = k/(x−a) + b has center (a, b)

③Intersection x = a and y = b = center of the hyperbola

④Domain: x ≠ a; range: y ≠ b

General Form → Standard Form

General Form

y = cx + dex + f

Polynomial division → quotient + remainder/divisor

🧮 Example: y = (2x+5)/(x+1)

①Divide: (2x+5) ÷ (x+1) = 2 r 3

②So y = 2 + 3/(x+1) = 3/(x+1) + 2

③Standard: k=3, a=−1, b=2

④Vertical x = −1, horizontal y = 2

⑤Hyperbola center: (−1, 2)

Conversion Tips

ListGeneral → Standard Conversion

Horizontal asymp. b

e.g. (2x+5)/(x+1) → b = 2/1 = 2

leading-coeff numerator ÷ denominator

Vertical asymp. a

e.g. x+1 = 0 → a = −1

x where denominator = 0

k value

e.g. 2x+5 = 2(x+1) + 3 → k = 3

remainder of numerator

Inverse of Rational Functions

Inverse Relation

y = kx − a + b ↔ y = kx − b + a

a and b swap — symmetric across y = x!

🪞 Self-Inverse?

①Inverse of y = k/x: x = k/y → y = k/x

②y = k/x is its own inverse (involution)

③General y = k/(x−a)+b: a and b swap

④If a = b, function is its own inverse (symmetric across y = x)

Work It Out

Example 1

Find the asymptotes of the rational function y = 2/(x − 3) + 1.

For y = k/(x − p) + q, the asymptotes are x = p and y = q.

y = 2/(x − 3) + 1

Match p = 3, q = 1.

Vertical asymptote x = 3, horizontal asymptote y = 1

▸ x = 3, y = 1

The x that makes the denominator 0 is the vertical asymptote; the constant term is the horizontal one.

Example 2

Rewrite y = (2x + 1)/(x − 1) in the form y = k/(x − p) + q and find its asymptotes.

Divide numerator by denominator: 2x + 1 = 2(x − 1) + 3.

y = (2(x − 1) + 3)/(x − 1) = 2 + 3/(x − 1)

Read the asymptotes from the standard form.

y = 3/(x − 1) + 2, asymptotes x = 1, y = 2

▸ x = 1, y = 2

For the general form, divide numerator by denominator to get standard form.

Wrap-up

Rational Function Core

y = kx − a + b

Asymptote intersection (a, b); domain x ≠ a; range y ≠ b

Grade-10 school exam type

The graph of y = 3/(x − 2) + 1 is symmetric about the point (a, b). What is a + b?

①1

②2

③3

④4

⑤5

▸ ③ 3

A rational function y = k/(x − p) + q is symmetric about the intersection of its asymptotes, (p, q).

Center of symmetry = (p, q) = (2, 1)

Add a = 2 and b = 1.

a + b = 2 + 1 = 3

🎯 Exam Points

①Asymptotes: x = a (vertical), y = b (horizontal) — denom=0 / leading-coeff ratio

②k > 0 in Q1·Q3; k < 0 in Q2·Q4

③(cx+d)/(ex+f) → standard via division

④Domain x ≠ a; range y ≠ b — asymptote values excluded

⑤Inverse: swap a and b; a=b → self-inverse (across y=x)

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Sets & Propositions

Irrational Functions

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