0580

Linear Graphs

Coordinate Geometry and Graphs

Reading a line from its equation

  • Any straight line that is not vertical can be written y = mx + c: the number m sets its steepness and the number c fixes where it meets the y-axis
  • The line therefore passes through (0, c)
    • y = 4x − 9 has gradient 4 and meets the y-axis at −9
    • y = 11 − 2x has gradient −2 and meets the y-axis at 11, so take the number attached to x, not the one written first
  • A horizontal line has the form y = c, since every point on it has the same y value
  • A vertical line has the form x = k, and it cannot be written as y = mx + c at all

Other forms of the equation

  • These papers also use the form ax + by = c, which has to be rearranged before the gradient can be read
  • Get a single y on one side, then divide every term by whatever multiplies it
    • 3x + 4y = 24 becomes 4y = −3x + 24, then y = −0.75x + 6, so the gradient is −0.75 and it crosses the y-axis at 6
  • Leaving 4y on the left and reading the gradient as −3 is the standard error here
  • Answers must be given fully simplified, so tidy fractions and collect any like terms before writing the final line
Exam tip

Get a single y on one side before reading anything off: in 4y = 3x + 8 the gradient is 0.75, not 3. "oe", meaning an equivalent form is accepted, appears in every one of the 43 mark schemes, so a correct rearrangement written differently still scores — but only once y is on its own.

Worked example

Reading a line from its equation

Write down the gradient and the y-axis crossing of y = 3x − 4, and of 2x + 5y = 20.

Solution:

  • The first is already in the form y = mx + c
  • The number in front of x is the gradient, so the gradient is 3
  • The constant is where the line crosses the y-axis, at (0, −4)
  • The second is not in that form, so rearrange it before reading anything
  • Subtract 2x: 5y = −2x + 20
  • Divide every term by 5: y = −0.4x + 4
  • The gradient is −0.4 and the line crosses the y-axis at (0, 4)
  • Reading a gradient of 2 straight off 2x + 5y = 20 is the standard slip: nothing can be read until y stands alone