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Cambridge IGCSE0607

Gradient of linear graphs

International Mathematics 0607 Chapter Notes

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Gradient of linear graphs
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In this chapter

  1. 1Gradient from a grid
  2. 2Gradient from two points
  3. 3Gradient from an equation
  4. 4Rearranging: divide every term
  5. 5Rearranging: two steps

1. Gradient from a grid

The gradient of a straight line tells you how steep it is. Pick two points on the line that sit exactly on grid crossings. Count the change in y (up or down) and the change in x (across). Read each change from the numbers on the axes.

A line that goes up to the right has a positive gradient. A line that goes down to the right has a negative gradient. A gradient can be a fraction, so never swap the division to avoid one.

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

Examiner insight

“Candidates made errors quoting the gradient formula or, if doing this by counting, forgot that this line has a negative gradient.” “Some tried to draw a triangle with AB as hypotenuse but when working out the gradient, decided it could not be a fraction so divided the wrong way.”

Worked example 12 marks

A straight line passes through the points A(1, 8) and B(5, 2) on the grid. Find the gradient of the line.

  1. 1

    change in y = 2 - 8 = -6 and change in x = 5 - 1 = 4 [1]

  2. 2

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

2. Gradient from two points

Given the coordinates of two points, subtract the y-values and subtract the x-values. The difference in y goes on top.

Subtract in the same order on the top and the bottom. Take care with negative coordinates: 5 - (-3) = 8.

m = (y₂ - y₁)/(x₂ - x₁)

Examiner insight

“Candidates did use the gradient formula and then made errors with the directed numbers.” “Only a few candidates wrote the formula the wrong way around with the xs on top.”

Worked example 12 marks

Do not use a calculator.

Find the gradient of the line through A(-3, 4) and B(5, -2).

  1. 1

    change in y = -2 - 4 = -6

  2. 2

    change in x = 5 - (-3) = 8 [1]

  3. 3

    gradient = (-6)/8 = -3/4 [1]

3. Gradient from an equation

When an equation is written as y equals something, the gradient is the coefficient of x. It is the number multiplying x, with its sign.

The number on its own is not the gradient. It is the y-intercept.

y = mx + c gradient = m

Examiner insight

“This tested understanding that the gradient is the coefficient of x rather than the first number after the equal sign as many candidates gave 7 as their answer.” “The instruction, ‘write down’ should have suggested to candidates that no calculation is necessary and all that they needed to do was to write down the coefficient of the x-term.”

Worked example 11 mark

Do not use a calculator.

Write down the gradient of the line y = 7 - 4x.

  1. 1

    the gradient is the coefficient of x, with its sign

  2. 2

    gradient = -4 [1]

4. Rearranging: divide every term

If the equation starts 3y = …, first get y on its own. Divide every term by the number in front of y.

Then read off the coefficient of x.

Examiner insight

“Many did not know how to find the gradient of the line from its equation. A small number gave 2x as the gradient.”

Worked example 12 marks

The equation of a line is 3y = 9 - 6x. Find the gradient of this line.

  1. 1

    divide every term by 3: y = 3 - 2x [1]

  2. 2

    gradient = -2 [1]

5. Rearranging: two steps

Sometimes x sits on the same side as y. Move it across first, changing its sign. Then divide every term by the number in front of y.

A gradient can be a fraction. Leave it as a fraction when the division does not come out exactly.

Examiner insight

“Only about half of the candidates could rearrange the equation correctly.”

Worked example 13 marks

The equation of a line is 4y - x = 12. Find the gradient and the y-intercept of the line.

  1. 1

    add x to both sides: 4y = x + 12

  2. 2

    divide every term by 4: y = 1/4x + 3 [1]

  3. 3

    gradient = 1/4 [1]

  4. 4

    y-intercept = 3 [1]

Recap

  • Gradient = change in y divided by change in x, read from the numbers on the axes.
  • Up to the right is positive. Down to the right is negative.
  • From two points: m = (y₂ - y₁)/(x₂ - x₁), subtracting in the same order on top and bottom.
  • In y = mx + c the gradient is the coefficient of x.
  • Rearrange first: get y on its own and divide every term.

End-of-chapter exercise

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

  1. A line goes down from left to right. State whether its gradient is positive or negative.1 mark
  2. The grid shows two straight lines, P and Q. Find the gradient of line P.2 marks
  3. Find the gradient of line Q on the grid in question 2.2 marks
  4. Do not use a calculator. Find the gradient of the line through (1, 2) and (4, 11).2 marks
  5. Do not use a calculator. Find the gradient of the line through P(-3, 4) and Q(2, -6).2 marks
  6. Do not use a calculator. Write down the gradient of the line y = 5 - 3x.1 mark
  7. Do not use a calculator. Write down the gradient of the line y = x/5 + 2.1 mark
  8. Do not use a calculator. The equation of a line is 5x + 2y = 8. Find the gradient of the line.2 marks
  9. A surveyor records three marker posts at A(-4.5, 7.2), B(1.5, -1.8) and C(5.5, -7.4). She says the three posts lie on one straight line. By calculating the gradients of AB and BC, decide whether she is correct.4 marks
  10. Two straight slides are drawn on a plan of a water park. Slide L is shown on the grid. Slide M lies along the line 4y + 9x = 180. (a) Find the gradient of L. (b) Find the gradient of M. (c) State which slide is steeper. Give a reason for your answer.5 marks

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