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9.3 Gradient

GCSE Maths Foundation · Edexcel · Chapter 9: Graphs

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: steepness of a line, finding the gradient from a graph or two points, and parallel lines. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Describe a gradient as positive, negative or zero
  • Find the gradient of a line from a graph using a right-angled triangle
  • Calculate the gradient between two points
  • Know that parallel lines have the same gradient

Key words

  • Gradient: How steep a line is: the number of units it goes up for every 1 unit across.
  • Rise: The change in y (up is positive, down is negative).
  • Run: The change in x, measured from left to right.
  • Parallel: Lines that go in the same direction and never meet.

What gradient means

The gradient measures steepness. Read every line from left to right. A line going uphill has a positive gradient. A line going downhill has a negative gradient. A horizontal line has gradient 0. The bigger the number (ignoring the sign), the steeper the line.

Key fact: Gradient = change in y ÷ change in x = rise ÷ run

Worked example: a ramp

A wheelchair ramp rises 0.5 m over a horizontal distance of 6 m. Find its gradient.
Gradient = 0.5 ÷ 6 = 1/12 (about 0.083).

Answer: 1/12

Gradient from a graph

Choose two points where the line crosses grid lines exactly. Draw a right-angled triangle under the line. Count the run (across) and the rise (up or down), using the scales on the axes.

Worked example: a positive gradient

From (0, 1) to (2, 5): rise = 4, run = 2. Gradient = 4 ÷ 2 = 2.

Answer: 2

Worked example: a negative gradient

From (0, 4) to (4, 2): the line goes down 2 while going 4 across. Gradient = −2 ÷ 4 = −1/2.

Answer: −1/2

Gradient from two points

You do not need a graph. Subtract the coordinates, keeping the points in the same order on the top and the bottom.

Key fact: Gradient between (x1, y1) and (x2, y2) = (y2 − y1) ÷ (x2 − x1)

Worked example: two points

Find the gradient of the line through (1, 3) and (4, 12).
Change in y = 12 − 3 = 9. Change in x = 4 − 1 = 3. Gradient = 9 ÷ 3 = 3.

Answer: 3

Worked example: with negatives

Find the gradient of the line through (−2, 5) and (2, −3).
Change in y = −3 − 5 = −8. Change in x = 2 − (−2) = 4. Gradient = −8 ÷ 4 = −2.
Check with a sketch: the line goes downhill, so a negative gradient makes sense.

Answer: −2

Parallel lines

Parallel lines are equally steep, so they have the same gradient. Lines with different gradients are not parallel.

Worked example: are they parallel?

Line A goes through (0, 2) and (2, 6). Line B goes through (1, −1) and (4, 5).
A: (6 − 2) ÷ (2 − 0) = 4 ÷ 2 = 2. B: (5 − (−1)) ÷ (4 − 1) = 6 ÷ 3 = 2.

Answer: both gradients are 2, so the lines are parallel

Worked example: using a gradient

A line has gradient 3 and passes through (2, 1). Find the y-coordinate of the point on the line where x = 6.
From x = 2 to x = 6 is 4 across. The line goes up 3 × 4 = 12, so y = 1 + 12 = 13.

Answer: (6, 13)

Common mistakes

  • Doing run ÷ rise instead of rise ÷ run. The change in y goes on top.
  • Losing the minus sign. A line going downhill from left to right has a negative gradient.
  • Counting squares when the two axes have different scales. Use the numbers on the axes.
  • Mixing the order of the points, e.g. (12 − 3) ÷ (1 − 4). Keep the same point first on top and bottom.

Revision questions

Try these, then check your answers below each one.

Question 1. Find the gradient of the line through (1, 2) and (4, 14).

Answer: 4

Working: (14 - 2) ÷ (4 - 1) = 12 ÷ 3 = 4.

Question 2. Find the gradient of the line through (−2, 7) and (2, −1).

Answer: -2

Working: (-1 - 7) ÷ (2 - (-2)) = -8 ÷ 4 = -2.

Question 3. Reading from left to right, a line goes down 3 squares for every 6 squares across. The scales on both axes are the same. What is its gradient? (Choose from: -1/2, 1/2, -2, 2.)

Answer: -1/2

Working: Rise = -3, run = 6: gradient = -3 ÷ 6 = -1/2.

Related lessons

Continue in the full course

This lesson is section 9.3 (EGCMF0903) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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