A reciprocal function is f(x) = 1x — the simplest function with asymptotes. A linear rational function is its bigger sibling, f(x) = ax + bcx + d. Both behave the same way — every key feature comes from the four constants.
📘
What you need to know
The reciprocal function y = 1x has asymptotes x = 0 and y = 0, no intercepts, and is self-inverse
A linear rational function y = ax + bcx + d has the same overall shape
Vertical asymptote at x = −dc (where the denominator is zero)
Horizontal asymptote at y = ac (the limiting value as |x| → ∞)
y-intercept at y = bd; x-intercept (root) at x = −ba
Sketches must label intercept coordinates and asymptote equations
The Reciprocal Function
The reciprocal function is defined as:
f(x) = 1x, x ≠ 0
Its domain is all real numbers except 0 (you can’t divide by zero), and its range is also all real numbers except 0 (the curve never actually equals zero). It’s a self-inverse function: applying it twice gets you back to where you started.
The Graph of y = 1/x
Key features of y = 1x
No y-intercept (the graph never touches the y-axis)
No roots (the graph never touches the x-axis)
Two asymptotes: horizontal y = 0 and vertical x = 0
Two axes of symmetry: y = x and y = −x
No maximum or minimum points
Self-inverse:f−1(x) = f(x)
Why asymptotes? As x gets close to 0, 1x blows up to ±∞ — that’s the vertical asymptote. As x gets very large, 1x shrinks to 0 — that’s the horizontal asymptote.
Linear Rational Functions
A linear rational function generalises the reciprocal — it’s a fraction with linear top and bottom:
f(x) = ax + bcx + d, x ≠ −dc
The reciprocal function 1x is just the special case where a = 0, b = 1, c = 1, d = 0. The four constants a, b, c, d tell you everything about the graph:
Reading the Constants
f(x) = ax + bcx + d
Horizontal Asymptote
y = ac
The limiting value of y as |x| → ∞.
Vertical Asymptote
x = −dc
The value that makes the denominator zero.
y-Intercept
(0, bd)
Substitute x = 0. Doesn’t exist if d = 0.
x-Intercept (root)
(−ba, 0)
Set the numerator = 0. Doesn’t exist if a = 0.
Anatomy of a Rational Graph
Here’s what a typical rational graph looks like — using f(x) = 10 − 5xx + 2 as an example (vertical asymptote at x = −2, horizontal at y = −5):
Linear Rational Graph — All Features Marked
The shape never changes. Every rational function looks like this — two branches separated by a vertical asymptote, both approaching a horizontal asymptote. Once you’ve located the four key features, the sketch is straightforward.
Domain and Range
For f(x) = ax + bcx + d:
Domain: all real numbers except x = −dc (the vertical asymptote)
Range: all real numbers except y = ac (the horizontal asymptote)
No turning points — the graph has no maximum or minimum.
Sketching Strategy
Find the vertical asymptote — set the denominator to zero and solve.
Find the horizontal asymptote — read off y = ac from the leading coefficients.
Find the y-intercept — substitute x = 0.
Find the x-intercept — set the numerator to zero and solve.
Draw the asymptotes as dashed lines.
Sketch the two branches through the intercepts, approaching the asymptotes but never crossing them.
Label everything — coordinates of intercepts and equations of asymptotes.
Worked Examples
✎
Example 1 — Reciprocal graph features
For f(x) = 1x, state: (a) the domain, (b) the range, (c) the equations of both asymptotes.
Answer:
(a) Domain: x can be any real number except 0 (denominator).x ∈ ℝ, x ≠ 0(b) Range: 1/x is never 0 (no value of x gives output 0).f(x) ∈ ℝ, f(x) ≠ 0(c) Asymptotes: where the function blows up or levels off.x = 0 and y = 0
✎
Example 2 — Find all features of a rational function
For f(x) = 3x + 6x − 2, find: (a) the asymptotes, (b) the intercepts.
Answer:
Identify a = 3, b = 6, c = 1, d = −2.(a) Vertical asymptote: denominator = 0.x − 2 = 0 → x = 2Horizontal asymptote: y = a/c.y = 3/1 = 3x = 2 and y = 3(b) y-intercept: substitute x = 0.f(0) = (0 + 6)/(0 − 2) = 6/(−2) = −3y-intercept: (0, −3)x-intercept: set numerator = 0.3x + 6 = 0 → x = −2x-intercept: (−2, 0)
✎
Example 3 — Full problem with sketch (asymptotes, intercepts, sketch)
The function f is defined by f(x) = 10 − 5xx + 2 for x ≠ −2.
(a) Write down the equations of the vertical and horizontal asymptotes.
Vertical: denominator = 0.x + 2 = 0x = −2Horizontal: limiting value as |x| → ∞.As x gets large, (10 − 5x)/(x + 2) ≈ −5x/x = −5y = −5Or just read off a/c = −5/1 = −5 (since a = −5, c = 1).
(b) Find the coordinates of the intercepts with the axes.
(c) Sketch the graph of f, labelling all key features.
Draw the asymptotes x = −2 and y = −5 as dashed lines.Plot the intercepts (0, 5) and (2, 0).Sketch two branches:Right branch (x > −2): passes through (0, 5) and (2, 0), approaches y = −5 from above as x → ∞.Left branch (x < −2): approaches y = −5 from below as x → −∞, dives to −∞ as x → −2 from the left.Always include both intercepts AND both asymptote equations on the sketch.
✎
Example 4 — Domain and range
For f(x) = 2x + 1x − 4, write down the domain and range.
Answer:
Domain: all real numbers except where the denominator is zero.x − 4 ≠ 0 → x ≠ 4x ∈ ℝ, x ≠ 4Range: all real numbers except the horizontal asymptote y = a/c.y = 2/1 = 2f(x) ∈ ℝ, f(x) ≠ 2A linear rational function never reaches its horizontal asymptote.
✎
Example 5 — Find the equation given features
A rational function has a vertical asymptote at x = 3, a horizontal asymptote at y = 2, and passes through the origin. Find an equation for f(x).
Answer:
Step 1: write the general form.f(x) = (ax + b) / (cx + d)Step 2: vertical asymptote at x = 3 means −d/c = 3.Pick c = 1, then d = −3 → denominator: x − 3Step 3: horizontal asymptote at y = 2 means a/c = 2.a/1 = 2 → a = 2Step 4: passes through origin (0, 0) → b/d = 0, so b = 0.f(x) = 2x / (x − 3)Check: f(0) = 0/(−3) = 0 ✓
💡
Tips
Always identify a, b, c, d first. Once you have all four, every key feature falls out from a simple ratio.
For a sketch, label everything: coordinates of both intercepts AND equations of both asymptotes. Missing labels = lost marks.
Use your GDC on Paper 2 to verify the shape, but be aware: most GDCs don’t draw the asymptotes. Add them yourself by reading the equation.
Asymptote shortcut: for ax + bcx + d, the horizontal asymptote is just the ratio of the leading coefficients a/c.
Check the y-intercept by substituting x = 0 — fastest way to verify the equation matches a sketch.
⚠
Common mistakes
Sign error in the vertical asymptote.x = −dc has a minus sign — for …x + 2, the asymptote is x = −2, not +2.
Forgetting to label the asymptotes on a sketch. Both vertical AND horizontal asymptote equations must be shown.
Drawing branches that cross the asymptotes. Asymptotes are barriers — the curve approaches but never touches them.
Using a/c for the y-intercept. The y-intercept is at b/d (set x = 0); a/c is the horizontal asymptote.
Including the asymptote in the range. The range excludes a/c — write f(x) ≠ a/c.
Believing your GDC’s sketch literally. GDCs often draw a connecting line through the vertical asymptote — that line isn’t real. Manually break the curve at the asymptote.
Final word: Identify a, b, c, d. Compute the four key features. Sketch with everything labelled. The shape is always the same — your job is just to place it correctly on the axes.
Need help with Functions?
Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.
