The idea, step by step
A constant product
Two non-zero quantities are inversely proportional when their product is constant. Writing y=k/x makes the relationship explicit: doubling x halves y, provided the same non-zero k is retained. Find k from one known pair by multiplying x and y; do not use the difference between the values.
Inverse proportion differs from a negative straight-line gradient. In y=k/x, equal additions to x do not cause equal changes in y. A useful diagnostic table includes the product xy: it should remain the same throughout a genuine inverse-proportion data set.
| x | y | xy |
|---|---|---|
| -6 | -4 | 24 |
| -3 | -8 | 24 |
| 2 | 12 | 24 |
| 4 | 6 | 24 |
| 8 | 3 | 24 |
Branches and asymptotes
For non-zero k, x=0 is excluded and y cannot be zero. The coordinate axes are asymptotes: the graph approaches them but does not meet them. For positive k, x and y have the same sign; for negative k, they have opposite signs. These facts locate the branches before you plot points.
Context can remove a branch. A fixed-distance travel model t=d/v uses positive speed and time, so only the first-quadrant branch is physically meaningful. A shifted reciprocal rule such as y=3/(x-2)+1 has shifted asymptotes; recognise the change instead of calling every reciprocal-shaped rule a direct inverse proportion between x and y.
A common trap
Thinking that a decreasing graph must show inverse proportion.
Instead: Check whether xy is constant. A decreasing line such as y=10-x has varying products and is not inverse proportion.
See the thinking
Find an inverse rule
y is inversely proportional to x, and y=8 when x=3. Find y when x=12.
- The constant is k=xy=24, so y=24/x.
- At x=12, y=2. The input has quadrupled, so the output is one quarter as large.
Travel time
A 180 km trip is travelled at constant speed v km/h. Write the travel time t, then compare 60 km/h and 90 km/h.
- Distance equals speed times time, so t=180/v for v>0.
- At 60 km/h, t=3; at 90 km/h, t=2.
- The negative branch of the mathematical graph does not represent this trip.
Check your understanding
If , what is the output at x=-5 and which inputs are excluded?
Show answer
; only x=0 is excluded from the algebraic domain.
30/(-5)=-6. Division by zero is undefined.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- y is inversely proportional to x, with y=6 at x=4. Find its rule.2 marks
- For , find y at x=3 and x=-6.2 marks
- Determine whether (2,12), (3,8), (6,4) fit one inverse-proportion rule.2 marks
- For , state its vertical and horizontal asymptotes.2 marks
- A rectangular garden has fixed area 48 m². Express width w in terms of length l, and find w when l=12 m.2 marks
- Explain why is not inverse proportion, even while it is positive and decreasing.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- y is inversely proportional to x, with y=6 at x=4. Find its rule.2 marks
The constant product is 4×6=24.
- For , find y at x=3 and x=-6.2 marks
Substitute each non-zero input separately.
-6 and 3. - Determine whether (2,12), (3,8), (6,4) fit one inverse-proportion rule.2 marks
Every pair has product 24.
Yes: . - For , state its vertical and horizontal asymptotes.2 marks
The denominator vanishes at -2. As the reciprocal term approaches zero, y approaches -3.
and . - A rectangular garden has fixed area 48 m². Express width w in terms of length l, and find w when l=12 m.2 marks
Use lw=48, then divide by positive l.
48/12=4.
, with l>0; w=4 m. - Explain why is not inverse proportion, even while it is positive and decreasing.2 marks
At x=1, y=10 and xy=10; at x=2, y=8 and xy=16.
Its product xy is not constant.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 Mathematics, Topic 1: Functions and graphs. Prerequisite algebra and coordinate geometry are embedded as supporting lessons.
Official curriculum source ↗