Year 11 Extended · Unit 4 Rational expressions and functions · Lesson 4 of 7

What does a graph look like when you divide by x?

What happens to k divided by x when x is very large, or very close to 0? This lesson draws the whole graph of y = k/x, reads its features from the rule, and uses it for inverse proportion.

FunctionsNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: if y is inversely proportional to x and y = 6 when x = 2, find y when x = 4.
    Answer3. The product xy stays 12.
  2. From Year 7: what is 15 ÷ 0.5?
    Answer30.
  3. From lesson 2: simplify (x2 − 9) ÷ (x − 3).
    Answerx + 3.
Words used in this lesson
asymptote
a line a graph approaches but never reaches. For example: a line y = 2 that a graph gets closer and closer to.
branch
one separate piece of a graph. For example: the part with x > 0.
domain
all the values of x that can be used. For example: every x ≥ 0 for y = √x.
range
all the values of y the graph reaches. For example: every y ≥ 0 for y = x2.
reciprocal
one divided by a number. For example: the reciprocal of 4 is 1/4.

What happens when you divide by something that gets smaller?

A tank in Kisumu holds 2400 litres and is emptied at x litres per minute, so the time it takes is T = 2400 ÷ x minutes. The pump speed x can be changed.

Predict first: what happens to the time as the pump speed gets larger and larger? And what happens as the speed gets closer and closer to 0? Then use the figure to check each prediction: y plays the part of T, and k is the size of the tank.

3
2
2the value of x
1.5the value of y

y = 3/x. At x = 2, y = 1.5; at x = −2, y = −1.5.

Show me the steps
  1. k = 3, x = 2: y = 1.5, and at x = −2, y = −1.5. The graph has two separate branches.
  2. Halve x down to 0.25: y = 12, above the top of the drawing. As x gets closer to 0, y grows without limit, so the y axis, x = 0, is a vertical asymptote.
  3. Move x out to 8: y = 0.375, close to the x axis but never 0. The x axis, y = 0, is a horizontal asymptote.
  4. Make k negative: y = −3/x. The branches swap to the second and fourth quadrants, and at x = 2, y = −1.5.

The graph of y = k/x has two branches and two asymptotes. An asymptote is a line that the graph gets as close to as you like but never reaches. Here they are the axes: x = 0 (vertical) and y = 0 (horizontal).

The domain is every x except 0 and the range is every y except 0. For positive k the branches are in the first and third quadrants, and for negative k in the second and fourth. The product xy is always k.

Two branches, one rule. Do not join the branches with a line through the origin: there is no point at x = 0, because k ÷ 0 has no value. And an asymptote is not a wall: it is a line the graph approaches. For this graph it is never touched.

Worked examples

1. For y = 8/x find y at x = 4, x = −2 and x = 0.5.

  1. Divide 8 by each: 8 ÷ 4 = 2, 8 ÷ (−2) = −4, 8 ÷ 0.5 = 16.
  2. State the domain and range: x ≠ 0 and y ≠ 0.

2. The graph of y = k/x passes through (5, 3). Find k, then find x when y = −6.

  1. k = xy = 5 × 3 = 15.
  2. −6 = 15/x gives x = 15 ÷ (−6) = −2.5.

3. Describe y = −12/x and find x when y = 3.

  1. k is negative, so the branches are in the second and fourth quadrants, with asymptotes x = 0 and y = 0.
  2. 3 = −12/x gives x = −4.

Your go. The first step is given. For y = 10/x, find y when x = −5 and find x when y = 0.4. The product xy is 10, so y = 10/x.

  1. Put x = −5 into 10/x.
  2. Put y = 0.4 into 0.4 = 10/x and solve.
Check the answer

y = 10 ÷ (−5) = −2, and x = 10 ÷ 0.4 = 25.

Check before you go on

1. For y = 5/x, which statement is true?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. y = 18/x. Find y when x = −4. Give your answer as a decimal.

3. y = 18/x. Find x when y = 0.75.

Level 2 · Building

4. The graph of y = k/x passes through the point (3, −8). Find k.

5. y = 24/x. By how much does y fall as x increases from 3 to 12?

Level 3 · Stretch

6. The graph of y = k/x passes through (2, 8) and through (n, 5). Find n. Give your answer as a decimal.

7. A lorry's journey takes t = 240/x hours at a steady speed of x km/h. The speed falls from 80 km/h to 60 km/h. How many more minutes does the journey take?

Something to think about

The graph of y = 1/x is unchanged when it is reflected in the line y = x. Why does that happen, and what does it say about its inverse function?

One way to think about it

Swapping x and y in y = 1/x gives x = 1/y, which is y = 1/x again. So the function is its own inverse: every point (a, 1/a) has a partner (1/a, a) on the same graph.

Exit ticket

8. The graph of y = k/x passes through (−2, 5). Which statement is correct?