Chapter 7 of 7

All 7 chapters
  1. 1Percentage increase and decrease, readunderstood
  2. 2Finding the original amount, readunderstood
  3. 3Simple and compound interest, readunderstood
  4. 4Growth and decay, readunderstood
  5. 5Harder ratio problems, readunderstood
  6. 6Proportion as a formula, readunderstood
  7. 7Graphs of proportion, readunderstood

M4.2.7 Chapter 7

Graphs of proportion

How can I tell from a graph that two amounts are in proportion?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Sketch the graphs of y = kx and y = k/x
  • Spot direct or inverse proportion from a table or graph

Ravi has a tap that fills 5 litres every minute. He plots minutes across and litres up. The rule is y = 5x.

Draw it first

Plot the tap

  1. A graph with the x axis time x (minutes) and the y axis water y (litres). Step 1 plots five dots: (1, 5), (2, 10), (3, 15), (4, 20) and (5, 25). Step 2 draws a straight line named y = 5x through them. Step 3 adds a dot at the start of the line, (0, 0).123456510152025300time x (minutes)water y (litres)

    1 Plot a dot for each minute: (1, 5), (2, 10), (3, 15), (4, 20), (5, 25).

  2. A graph with the x axis time x (minutes) and the y axis water y (litres). Step 1 plots five dots: (1, 5), (2, 10), (3, 15), (4, 20) and (5, 25). Step 2 draws a straight line named y = 5x through them. Step 3 adds a dot at the start of the line, (0, 0).y = 5x123456510152025300time x (minutes)water y (litres)

    2 The dots lie in a straight line. Draw the line through them.

  3. A graph with the x axis time x (minutes) and the y axis water y (litres). Step 1 plots five dots: (1, 5), (2, 10), (3, 15), (4, 20) and (5, 25). Step 2 draws a straight line named y = 5x through them. Step 3 adds a dot at the start of the line, (0, 0).y = 5x123456510152025300time x (minutes)water y (litres)

    3 At 0 minutes there is 0 litres, so the line starts at (0, 0), the origin.

The graph of y = kx is always a straight line through the origin, the point (0, 0). The number k says how steep the line is. It is the gradient: how much y goes up for each 1 across. Here k = 5, so y goes up 5 for each 1 across.

A graph with the x axis x and the y axis y. Three straight lines all start at the origin: y = 0.5x, y = x, y = 2x. The bigger the number in front of x, the steeper the line.y = 0.5xy = xy = 2x1234561234560xy
A bigger k gives a steeper line. All three lines pass through the origin.

Worked example

Sketch the graph of y = 3x for x from 0 to 4.

  1. Choose some values of x: 0, 1, 2, 3 and 4.
  2. Work out y = 3x for each: y = 0, 3, 6, 9 and 12. The last is y = 12.
  3. Plot the points (0, 0), (1, 3), (2, 6), (3, 9) and (4, 12).
  4. Join them with a straight line. It goes through the origin.

Answer: A straight line through (0, 0) and (4, 12)

A graph with the x axis x and the y axis y. Dots at (0, 0), (1, 3), (2, 6), (3, 9) and (4, 12) lie on a straight line named y = 3x.y = 3x1234246810120xy
The graph of y = 3x.

A straight line is not enough. Compare y = 2x with y = 2x + 3. The second line is straight, but it crosses the y axis at 3, not at 0. When x = 0 it gives y = 3. Doubling x does not double y, so it is not proportion.

A graph with the x axis x and the y axis y. A line named y = 2x starts at the origin. A parallel line named y = 2x + 3 starts higher, with a dot at (0, 3) where it crosses the y axis.y = 2xy = 2x + 312345636912150xy
Both lines are straight. Only the one through the origin is proportion.

Now inverse proportion. Take y = 12 ÷ x, for x above 0. When x is 1, 2, 3, 4, y is 12, 6, 4, 3. The points do not lie in a line. They lie on a curve that falls.

See it three ways

With things

Rectangles that hold 12 squares: 1 across and 12 up, 2 across and 6 up, 3 across and 4 up.

As a picture

A graph with the x axis x and the y axis y. Dots at (1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1) lie on a falling curve named y = 12 ÷ x. The curve never touches either axis.y = 12 ÷ x24681012246810120xy

In symbols

x × y = 12, so y = 12 ÷ x.

The curve gets very high near the y axis, and very close to the x axis far along. It comes as near as you like to each axis but never touches either. A y of 0 would need 12 ÷ x = 0, and no x gives that.

Worked example

Sketch the graph of y = 12 ÷ x for x from 1 to 12.

  1. Choose values of x that divide 12: 1, 2, 3, 4, 6 and 12.
  2. When x = 2, y = 12 ÷ 2 = 6.
  3. When x = 4, y = 12 ÷ 4 = 3.
  4. When x = 6, y = 12 ÷ 6 = 2.
  5. Plot the points and join them with a smooth curve. It falls and never touches the axes.

Answer: A smooth curve through (2, 6), (4, 3) and (6, 2)

You can also tell the two kinds apart from a table. For direct proportion, y ÷ x is the same every time. For inverse proportion, x × y is the same every time.

A table with the head x, y, y ÷ x, x × y. Rows: 2, 3, 1.5, 6; 4, 6, 1.5, 24; 6, 9, 1.5, 54; 8, 12, 1.5, 96.xyy ÷ xx × y231.56461.524691.5548121.596
Test a table: divide, then multiply. One column stays the same.

Worked example

A table has x = 2, 4, 6, 8 and y = 3, 6, 9, 12. Is y proportional to x, inversely proportional, or neither?

  1. Divide y by x for the first pair: 3 ÷ 2 = 1.5.
  2. Divide for the others: 6 ÷ 4, 9 ÷ 6 and 12 ÷ 8 are all 1.5.
  3. Now multiply: 2 × 3 = 6, but 4 × 6 = 24. This is not fixed.
  4. y ÷ x is always 1.5, so y is proportional to x, with k = 1.5.

Answer: Directly proportional, with k = 1.5

Try it

Plot y = 2x on squared paper

You need: squared paper, a ruler, a pencil.

  1. Draw an x axis from 0 to 4 and a y axis from 0 to 8.
  2. Plot the points (0, 0), (1, 2), (2, 4), (3, 6) and (4, 8).
  3. Lay the ruler along the points and draw the line.

What you will see: The points lie on one straight line, and the line passes through the origin. Each step across goes up 2.

If you cannot try it, this is what happens.
A graph with the x axis x and the y axis y. Dots at (0, 0), (1, 2), (2, 4), (3, 6) and (4, 8) lie on a straight line named y = 2x.y = 2x123424680xy

A wrong idea many people have

Any straight line shows direct proportion.

Why it feels right: Direct proportion makes a straight line, so a straight line feels like proof.

The test that breaks it: Take y = 2x + 3. When x = 1, y = 5. When x = 2, which is double, y = 7. That is not double 5.

What is true: Direct proportion needs a straight line through the origin. Zero of x must give zero of y.

A wrong idea many people have

If y goes down as x goes up, the graph shows inverse proportion.

Why it feels right: Inverse proportion does go down as x goes up, so any fall looks like it.

The test that breaks it: Take y = 10 − x. It falls as x goes up. When x = 2, y = 8, and x × y = 16. When x = 4, y = 6, and x × y = 24. The product is not fixed.

What is true: Inverse proportion needs x × y to stay the same. Its graph is a curve that never touches the axes, not a straight line.

Key words

origin
the point (0, 0), where the two axes cross.
gradient
how steep a line is: how much y goes up for each 1 across.

Back to your guess

The answer is B: The graph is a straight line that goes through the origin.

Direct proportion makes a straight line through the origin, the point where both amounts are zero. A straight line that misses the origin is not proportion. A curve that falls and never touches the axes shows inverse proportion.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which description fits the graph of y = kx, with k above 0?

    Tap your answer.

    1. Not this one. A straight line that misses the origin, such as y = 2x + 3, is not proportion. When x is 0, y must be 0. Go to M4.2.7. Try another.

    2. Not this one. That is the graph of y = k ÷ x, inverse proportion. The graph of y = kx rises in a straight line. Go to M4.2.7. Try another.

    3. Yes, that is right. When x = 0, y = k × 0 = 0, so the line passes through the origin. All the points are in a straight line because y goes up by k for each 1 across.

    4. Not this one. k is the steepness of the line, not where it crosses the y axis. The line y = kx crosses the y axis at 0. Go to M4.2.7. Try another.

    See the answer

    A straight line through the origin

    When x = 0, y = k × 0 = 0, so the line passes through the origin. All the points are in a straight line because y goes up by k for each 1 across.

    • If you chose A: A straight line that misses the origin, such as y = 2x + 3, is not proportion. When x is 0, y must be 0. Go to M4.2.7.
    • If you chose B: That is the graph of y = k ÷ x, inverse proportion. The graph of y = kx rises in a straight line. Go to M4.2.7.
    • If you chose D: k is the steepness of the line, not where it crosses the y axis. The line y = kx crosses the y axis at 0. Go to M4.2.7.

  2. The table shows x and y. Which is it?

    A table with the head x, y. Rows: 2, 8; 3, 12; 5, 20; 6, 24.xy28312520624
    Tap your answer.

    1. Yes, that is right. y ÷ x is 8 ÷ 2 = 4 for every pair, so y = 4x.

    2. Not this one. In inverse proportion y goes down when x goes up. Here y goes up with x, so it is not inverse. Go to M4.2.6. Try another.

    3. Not this one. Divide y by x for each pair: 8 ÷ 2, 12 ÷ 3, 20 ÷ 5 and 24 ÷ 6 are all 4. A fixed answer means direct proportion. Go to M4.2.6. Try another.

    See the answer

    Direct proportion, y = 4x

    y ÷ x is 8 ÷ 2 = 4 for every pair, so y = 4x.

    • If you chose B: In inverse proportion y goes down when x goes up. Here y goes up with x, so it is not inverse. Go to M4.2.6.
    • If you chose C: Divide y by x for each pair: 8 ÷ 2, 12 ÷ 3, 20 ÷ 5 and 24 ÷ 6 are all 4. A fixed answer means direct proportion. Go to M4.2.6.

  3. The table shows x and y. Which is it?

    A table with the head x, y. Rows: 1, 16; 2, 8; 4, 4; 8, 2.xy116284482
    Tap your answer.

    1. Not this one. In direct proportion y goes up when x goes up. Here y goes down as x goes up. Go to M4.2.6. Try another.

    2. Yes, that is right. x × y is 1 × 16 = 16 for every pair, so y = 16 ÷ x.

    3. Not this one. Multiply x by y for each pair: 1 × 16, 2 × 8, 4 × 4 and 8 × 2 are all 16. A fixed answer means inverse proportion. Go to M4.2.6. Try another.

    See the answer

    Inverse proportion, y = 16 ÷ x

    x × y is 1 × 16 = 16 for every pair, so y = 16 ÷ x.

    • If you chose A: In direct proportion y goes up when x goes up. Here y goes down as x goes up. Go to M4.2.6.
    • If you chose C: Multiply x by y for each pair: 1 × 16, 2 × 8, 4 × 4 and 8 × 2 are all 16. A fixed answer means inverse proportion. Go to M4.2.6.

  4. The table shows x and y. Which is it?

    A table with the head x, y. Rows: 1, 4; 2, 6; 3, 8; 4, 10.xy142638410
    Tap your answer.

    1. Not this one. Divide y by x: 4 ÷ 1 = 4, but 6 ÷ 2 = 3. The answer is not fixed. Also, when x = 0 this pattern would give y = 2, not 0. Go to M4.2.6. Try another.

    2. Not this one. In inverse proportion y goes down as x goes up. Here y goes up with x, and x × y is 4, 12, 24 and 40. Go to M4.2.6. Try another.

    3. Yes, that is right. Dividing y by x gives 4, 3, 2.67 and 2.5, which is not fixed. The product x × y is 4, 12, 24 and 40, which is not fixed either. The pattern is y = 2x + 2, a straight line that misses the origin.

    See the answer

    Neither

    Dividing y by x gives 4, 3, 2.67 and 2.5, which is not fixed. The product x × y is 4, 12, 24 and 40, which is not fixed either. The pattern is y = 2x + 2, a straight line that misses the origin.

    • If you chose A: Divide y by x: 4 ÷ 1 = 4, but 6 ÷ 2 = 3. The answer is not fixed. Also, when x = 0 this pattern would give y = 2, not 0. Go to M4.2.6.
    • If you chose B: In inverse proportion y goes down as x goes up. Here y goes up with x, and x × y is 4, 12, 24 and 40. Go to M4.2.6.

  5. y is proportional to x. The graph is a straight line through the origin and through the point A (4, 10). What is k?

    A graph with the x axis x and the y axis y. A straight line named y = kx starts at the origin. A dot named A (4, 10) is on the line, with dashed lines down to the x axis and across to the y axis.A (4, 10)y = kx12345636912150xy
    Tap your answer.

    1. Not this one. You divided x by y. The number k is y ÷ x, which is 10 ÷ 4. Go to M4.2.6. Try another.

    2. Not this one. You took 4 away from 10. Proportion divides: k = y ÷ x. Go to M4.2.6. Try another.

    3. Not this one. 10 is the y value of A. k is how much y goes up for each 1 across, which is 10 ÷ 4. Go to M4.2.6. Try another.

    4. Yes, that is right. k = y ÷ x = 10 ÷ 4 = 2.5. For every 1 across, the line goes up 2.5.

    See the answer

    2.5

    k = y ÷ x = 10 ÷ 4 = 2.5. For every 1 across, the line goes up 2.5.

    • If you chose A: You divided x by y. The number k is y ÷ x, which is 10 ÷ 4. Go to M4.2.6.
    • If you chose B: You took 4 away from 10. Proportion divides: k = y ÷ x. Go to M4.2.6.
    • If you chose C: 10 is the y value of A. k is how much y goes up for each 1 across, which is 10 ÷ 4. Go to M4.2.6.

  6. The curve shows inverse proportion. It passes through (2, 6), (3, 4) and (4, 3). Which formula is right?

    A graph with the x axis x and the y axis y. A falling curve passes through three dots: (2, 6), (3, 4), (4, 3).2468246810120xy
    Tap your answer.

    1. Yes, that is right. x × y is 2 × 6 = 12, 3 × 4 = 12, 4 × 3 = 12. The product is always 12, so y = 12 ÷ x.

    2. Not this one. This fits (2, 6), but when x = 3 it gives 9, not 4. It is also a rising line, not a falling curve. Go to M4.2.6. Try another.

    3. Not this one. This fits (2, 6), but when x = 3 it gives 5, not 4. A straight line is not inverse proportion. Check x × y for each point. Go to M4.2.6. Try another.

    See the answer

    y = 12 ÷ x

    x × y is 2 × 6 = 12, 3 × 4 = 12, 4 × 3 = 12. The product is always 12, so y = 12 ÷ x.

    • If you chose B: This fits (2, 6), but when x = 3 it gives 9, not 4. It is also a rising line, not a falling curve. Go to M4.2.6.
    • If you chose C: This fits (2, 6), but when x = 3 it gives 5, not 4. A straight line is not inverse proportion. Check x × y for each point. Go to M4.2.6.

  7. From an earlier chapter

    y is inversely proportional to x. When x = 5, y = 6. Find y when x = 3.

    See the answer

    10

    k = x × y = 5 × 6 = 30. Then y = 30 ÷ 3 = 10.

  8. y is proportional to x. The graph is a straight line through the origin and through the point (6, 21). What is k?

    See the answer

    3.5

    k = y ÷ x = 21 ÷ 6 = 3.5.

Keep it

Copy this into your exercise book.

The graph of y = kx is a straight line through the origin. The graph of y = k ÷ x is a curve that never touches the axes. A table: y ÷ x fixed is direct, x × y fixed is inverse.

Draw: A straight line through the origin beside a falling curve, with y ÷ x and x × y written under them.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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