0580

Linear Graphs

Coordinate Geometry and Graphs

From an equation

  • Build a small table of values by substituting three x values that are spread across the axes you are given
  • Three points, not two: if all three line up, none of them is wrong, whereas two points always look convincing
  • Plot the points and join them with a ruled line that runs the full width of the grid
Two straight lines drawn on the same grid: a steep uphill line through (0, −1) and (1, 2), stepping 1 across and 3 up, and a shallow downhill line through (0, 3) and (5, 0)
Source: Drawing linear graphs by Save My Exams
  • Alternatively start at (0, c) and step out the gradient, going one right and m up each time, down instead if m is negative
  • For a gradient written as a fraction, step across by the bottom number and up by the top number
A shallow uphill line on a grid through (0, −1) and (3, 1), with a dashed triangle showing 3 across against 2 up, giving a gradient of two thirds
Source: Gradient of a line by Save My Exams

From the ax + by = c form

  • A quicker route for this form is to find where the line meets each axis
  • Substitute x = 0 to get the point on the y-axis, then y = 0 to get the point on the x-axis
  • Join those two points and extend the line across the grid
    • For 3x + 4y = 24, setting x = 0 gives (0, 6) and setting y = 0 gives (8, 0)
Exam tip

Plot three points, not two. Two points always lie on a line, so a slip in either goes unnoticed; a third that misses the line tells you at once which one is wrong. "Draw the graph of" is asked in 16 of the 43 papers and the line must be ruled.

Worked example

Drawing a line two ways

Draw y = 2x + 1 for x from −1 to 2, and draw 3x + 2y = 12.

Solution:

  • For the first, build a short table of values
  • x = −1 gives y = −1; x = 0 gives 1; x = 1 gives 3; x = 2 gives 5
  • Plot (−1, −1), (0, 1), (1, 3) and (2, 5), and join them with a ruled line
  • For the second, the cover-up method is quicker than rearranging
  • Setting x = 0 gives 2y = 12, so y = 6 and the line passes through (0, 6)
  • Setting y = 0 gives 3x = 12, so x = 4 and the line passes through (4, 0)
  • Plot those two intercepts and rule the line through them
  • Two points fix a straight line, but a third is worth plotting as a check