A hyperbola in two branches: The reciprocal function y = 1/x is a hyperbola with two separate branches — one in the top-right, one in the bottom-left. As x grows, y shrinks toward 0; as x nears 0, y shoots off to ±∞.
It never touches the axes: 1/x is never 0 (no x-intercept) and is undefined at x = 0 (no y-intercept). The axes are its asymptotes.
Big in, small out: Large x → tiny y; tiny x → huge y. The two are reciprocals, so the curve hugs both axes.
Animated graph
Watch the graph build step by step in study mode.
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it free for 7 days:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
x ≠ 0, y ≠ 0: For y = 1/x: the domain is x ≠ 0 (can't divide by zero) and the range is y ≠ 0 (the output is never 0). The asymptotes are the lines x = 0 (vertical) and y = 0 (horizontal).
IB-style question — state the features
State the domain, range and asymptotes of y = 1/x.
Step by step
- Denominator zero is banned.
- Output never reaches 0.
Final answer
Domain x ≠ 0, range y ≠ 0, asymptotes x = 0 and y = 0.
Animated graph
Watch the graph build step by step in study mode.
Asymptotes are dashed guides: Draw the asymptotes first (here the two axes); the branches then hug them.
Study smarter, not longer
Most students waste 40% of study time on topics they already know. Our AI tracks your progress and optimizes every minute.
The asymptotes move with the shift: y = 1/(x − h) + k is y = 1/x slid h right and k up. So the vertical asymptote becomes x = h and the horizontal asymptote becomes y = k.
IB-style question — shifted reciprocal
State the asymptotes of y = 1/(x − 3) + 2.
Step by step
- Vertical: denominator zero.
- Horizontal: the +2 raises the level.
Final answer
Vertical asymptote x = 3, horizontal asymptote y = 2.
Animated graph
Watch the graph build step by step in study mode.
Sign of h: 1/(x − 3) shifts right 3 (vertical asymptote x = +3); 1/(x + 3) shifts left (x = −3).
Asymptotes, then the two branches: To sketch: draw the asymptotes (dashed), find any intercepts (set x = 0 for the y-intercept, set y = 0 for the x-intercept), then draw the two branches hugging the asymptotes.
IB-style question — intercepts of a shifted reciprocal
Find the y-intercept of y = 1/(x − 3) + 2.
Step by step
- Set x = 0.
- Simplify.
Final answer
y-intercept at (0, 5/3).
Animated graph
Watch the graph build step by step in study mode.
Reciprocals have at most one of each intercept: A shifted reciprocal crosses each axis at most once (sometimes not at all if an asymptote is in the way).