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Cambridge IGCSE0580

Gradient of linear graphs

Mathematics 0580 Chapter Notes

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Gradient of linear graphs
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In this chapter

  1. 1Gradient is rise over run
  2. 2Positive and negative gradients
  3. 3Gradient from two points
  4. 4Negative coordinates

1. Gradient is rise over run

The gradient of a straight line measures how steep it is. It is the same everywhere on the line.

On a grid, pick two points where the line crosses grid corners. The vertical distance between them is the rise. The horizontal distance is the run.

Divide the rise by the run. Always put the vertical change on top.

gradient = rise/run = (change in y)/(change in x)

Examiner insight

“Probably more successful was the method of forming a triangle on the diagram to work out rise ÷ run, especially as the scales were the same.”

Worked example 12 marks

A straight line on a grid passes through the points (1, 2) and (5, 5). Find the gradient of the line.

  1. 1

    rise = 5 - 2 = 3 and run = 5 - 1 = 4 [1]

  2. 2

    gradient = rise/run = 3/4 [1]

2. Positive and negative gradients

A line that slopes up from left to right has a positive gradient. A line that slopes down from left to right has a negative gradient.

Check the sign against the picture. A downhill line can never have a positive gradient.

Examiner insight

“A negative gradient which was also a fraction was found challenging by all but the most able candidates.” “A check of the diagram shows that the gradient is negative and that a positive intercept is impossible.”

Worked example 12 marks

Do not use a calculator. A straight line on a grid passes through the points (0, 4) and (6, 0). Find the gradient of the line.

  1. 1

    the line slopes down, so the gradient is negative

  2. 2

    rise = 0 - 4 = -4 and run = 6 - 0 = 6 [1]

  3. 3

    gradient = (-4)/6 = -2/3 [1]

3. Gradient from two points

When the coordinates of two points are given, no grid is needed. Call the points (x₁, y₁) and (x₂, y₂).

Subtract the y-coordinates for the change in y. Subtract the x-coordinates in the same order for the change in x. Then divide.

gradient = (y₂ - y₁)/(x₂ - x₁)

Examiner insight

“A significant number of candidates incorrectly used the co-ordinates (3, 0) instead of (0, 3) in their gradient calculation.”

Worked example 12 marks

The coordinates of A are (3, 20). The coordinates of B are (5, 10). Work out the gradient of AB.

  1. 1

    gradient = (10 - 20)/(5 - 3) [1]

  2. 2

    = (-10)/2 = -5 [1]

4. Negative coordinates

Subtracting a negative number adds. Write each subtraction in brackets so the signs are clear.

Then check the sign of your answer against the points. If the line goes down as x increases, the gradient must be negative.

Examiner insight

“The negative values often led to mistakes in both of these steps.”

Worked example 12 marks

Find the gradient of the line through the points (-2, 6) and (4, -3).

  1. 1

    gradient = (-3 - 6)/(4 - (-2)) [1]

  2. 2

    = (-9)/6 = -3/2 [1]

Recap

  • Gradient measures steepness. It is the same everywhere on a straight line.
  • Gradient = rise ÷ run = change in y ÷ change in x.
  • Slopes up: positive. Slopes down: negative.
  • Use the same order of subtraction for y and for x.
  • Extended: gradient from two points = (y₂ - y₁)/(x₂ - x₁).
  • Subtracting a negative number adds.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. State whether the gradient is positive or negative for a line that slopes down from left to right.1 mark
  2. Do not use a calculator. On a grid, a straight line rises 6 squares for every 2 squares across. Find its gradient.2 marks
  3. Do not use a calculator. A straight line on a grid passes through the points (0, 5) and (4, 0). Find its gradient.2 marks
  4. A path is drawn as a straight line on a grid. It rises 40 m for every 160 m across. Write its gradient as a decimal.2 marks
  5. A straight line on a grid passes through the points (-3, -1) and (3, 3). Find its gradient as a fraction in its simplest form.2 marks
  6. Do not use a calculator. Find the gradient of the line through A(3, 1) and B(7, 9).2 marks
  7. Find the gradient of the line through P(-2, 5) and Q(4, -1).2 marks
  8. Find the gradient of the line through A(-4, -3) and B(2, 6). Give your answer as a decimal.2 marks
  9. A student finds the gradient of the line through A(-5, 4) and B(-1, -8). The student writes (-8 - 4)/(-1 - 5) = (-12)/(-6) = 2. Describe the mistake and find the correct gradient.3 marks
  10. Do not use a calculator. A straight pipe is drawn on a coordinate grid. It passes through A(-7, 6) and B(-1, 2). Two more points are marked at C(8, -4) and D(11, -5). One of them lies on the line of the pipe. Use gradients to decide which one.5 marks

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