Inverse proportion as a function · 跟马丁学编程

The more, the less

You have met inverse proportion since year 7: the more painters, the shorter the job. Today you look at it with new eyes — as a function.

The rule is y = k/x. The number k is fixed, x is the input. For example y = 12/x: for x = 1, y = 12, for x = 2, y = 6, for x = 3, y = 4, for x = 6, y = 2.

See the rule? The product x · y is always the same — always 12. You double the input, the output shrinks to a half. That is why “inverse”: the quantities pull in opposite directions.

The graph is neither a straight line nor a parabola. It is a hyperbola — a curve that gets close to the axes but never touches them. It cannot: for x = 0 the value does not exist (dividing by zero — remember the conditions from lesson 1) and y = 0 would want infinitely large x.

Today you will draw a hyperbola and learn to recognise inverse proportion among other functions.

Six friends ordered a pizza party for 720 Kč. “How much each?” Filip asks. “720 divided by six, 120 a head,” Tereza counts. But two do not arrive. “So 720 divided by four — 180.” Filip whistles: “The fewer of us, the more we pay. That is inverse proportion: y = 720/x.” Tereza draws a table: 2 people at 360, 3 at 240, 4 at 180, 6 at 120. “The product is always 720. The bill does not change, it just gets sliced differently.”

💡

The recognition sign of inverse proportion: the product x · y is the same for all pairs. When you find a constant product in a table, you have y = k/x and that k is exactly the product.

A hyperbola from a table

Step 1: A table for y = 12/x. I pick divisors of twelve, so the numbers are nice:

x: 1, 2, 3, 4, 6, 12 y: 12, 6, 4, 3, 2, 1

Step 2: Check with the product. 1 · 12 = 12, 2 · 6 = 12, 3 · 4 = 12… Everywhere 12. That is k.

Step 3: Plot and join with a smooth curve. No ruler — the points lie on an arch that from the left falls steeply down and to the right slowly creeps towards the x-axis. It gets close to the axes, it never crosses them.

Why it does not touch the axes: x = 0 is forbidden (12/0 does not exist — condition x ≠ 0). And y = 0 would mean 12/x is zero — that does not happen for any x.

How to tell inverse proportion from the others:

y = 3x — direct proportion, a straight line from the origin. Twice as big x, twice as big y. y = 12/x — inverse proportion, a hyperbola. Twice as big x, half y. y = x² — a parabola. Neither of those.

In practice: a bill split between people, a distance divided by speed (travel time), work divided by the number of workers. Whenever a fixed whole is split, y = k/x is in play.

Example 1: a table and k

Question: The function y = 24/x. Build a table for x = 1, 2, 3, 4, 6, 8 and check the constant product.

Method: I divide: 24/1 = 24, 24/2 = 12, 24/3 = 8, 24/4 = 6, 24/6 = 4, 24/8 = 3.

Products: 1 · 24 = 24, 2 · 12 = 24, 3 · 8 = 24, 4 · 6 = 24… everywhere 24.

Answer: The table falls, but not evenly: between x = 1 and x = 2, y drops by 12, between x = 6 and x = 8 only by 1. That is exactly how a hyperbola looks — steeply down, then gently.

Example 2: a trip

Question: To camp it is 120 km. Write the travel time as a function of speed and calculate the time for 40, 60 and 80 km/h.

Method: Time = distance/speed, so y = 120/x (x in km/h, y in hours).

For x = 40: y = 120/40 = 3 hours. For x = 60: y = 120/60 = 2 hours. For x = 80: y = 120/80 = 1.5 hours.

Product check: 40 · 3 = 120, 60 · 2 = 120, 80 · 1.5 = 120. It matches.

Answer: The faster, the shorter the time — but speeding up from 60 to 80 saves only half an hour, while from 40 to 60 a whole hour.

Example 3: find k from one pair

Question: The quantities x and y are inversely proportional. For x = 4, y = 9. Write the rule and calculate y for x = 6.

Method: For inverse proportion, k = x · y. So k = 4 · 9 = 36.

Rule: y = 36/x.

For x = 6: y = 36/6 = 6.

Check: 6 · 6 = 36 — the product holds.

Answer: y = 36/x and for x = 6, y = 6. One pair of values is enough for the whole rule — that is the power of a constant product.

⚠️

Not every falling is inverse proportion. The function y = −2x + 10 falls too, but it is a straight line and the product x · y changes. Inverse proportion has two recognition signs at once: the rule y = k/x and a constant product in the table. Check the product for two or three pairs before you call a problem inverse proportion.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

Practise substituting into expressions — you calculate hyperbola values with the same move. Write a whole number.

Practise the questions

On paper

One hyperbola, one decision and one calculation from life.

Build a table of y = 18/x for x = 1, 2, 3, 6, 9, 18 and draw the graph with a smooth curve. Check the constant product.
Decide with a reason: the problem “5 notebooks cost 60 Kč, how much do 8 notebooks cost” — direct or inverse proportion? And the problem “3 pumps empty a pool in 8 hours, how long do 4 pumps take”?
A job for 960 Kč is split among a group. Calculate the share per person for 2, 4, 6 and 8 people.
The quantities are inversely proportional and for x = 3, y = 8. Find k, write the rule and calculate y for x = 12.

Now you 💪

From one pair of values you can find k (by the product) and write the rule y = k/x.
You tell direct and inverse proportion apart by whether the quantities grow together or against each other.
You know why a hyperbola never crosses the axes.

✅

Done when: You have drawn a hyperbola with a constant product, correctly sorted problems into direct and inverse proportion, and one rule found from a single pair.

What to take from this lesson

Inverse proportion is the function y = k/x. The graph is a hyperbola.
Recognition sign: the product x · y is the same for all pairs (it equals k).
Doubling x means half y.
A hyperbola gets close to the axes but never crosses them — x = 0 is forbidden.
A fixed whole split among more parts = inverse proportion in practice.