Algebra - Graphs: plot graphs of equations that correspond to straight-line graphs in the coordinate plane; use the form y = mx + c to identify parallel lines; find the equation of the line through two given points or through one point with a given gradient (Foundation tier)

The idea

To draw a line from its equation, build a short table of values: choose three x values, work out y for each, plot them and join them with a ruler. For y = 2x − 1, x = 0 gives y = −1, x = 1 gives y = 1, and x = 3 gives y = 5. If the equation arrives as 2y + 4x = 8, rearrange it to y = −2x + 4 before you start. Written that way the number in front of x is the gradient and the number on its own is where the line crosses the y axis. Two different lines with the same gradient never meet, so they are parallel. Given two points, the gradient is the change in y divided by the change in x: through (1, 4) and (3, 10) that is 6/2 = 3, and putting (1, 4) back in gives 4 = 3 + c, so the line is y = 3x + 1. Given one point and a gradient, substitute the point to find c: through (2, 7) with gradient 3, 7 = 3 × 2 + c gives c = 1, so y = 3x + 1.

What earns the marks

• A correct table of values, or for three correct points, even if the line is then drawn badly.
• One straight ruled line through the points, drawn across the range of x the question gives.
• Rearranging an equation so that y is on its own before the gradient is read off.
• The gradient as a change in y over a change in x, even if the division slips.
• Substituting a point to find the intercept, and one for the finished equation.
• Using a given gradient with a given point to find c, as in 7 = 3 × 2 + c giving c = 1.

Common mistakes and how to put them right

• Working out the gradient as the change in x over the change in y.

  Put it right: It is the change in y divided by the change in x.

• Giving a positive gradient for a line that falls from left to right.

  Put it right: A falling line has a negative gradient.

• Joining the plotted points with several short segments instead of one ruled line.

• Deciding lines are parallel from the intercepts rather than the gradients, so treating y = 3x + 2 and y = 5x + 2 as parallel.

  Put it right: Equal gradients, not equal intercepts, make lines parallel.

• Substituting the given point into the wrong place when finding c, such as swapping x and y.

Exam tips

• Draw the line right across the grid the question gives, because a short line can miss the points an examiner checks.
• Put x = 0 in your table of values, since it hands you the y intercept with no extra work.
• When two points are given, mark the change in y and the change in x on the diagram before you divide.
• When one point and a gradient are given, write y = mx + c with m filled in, then substitute the point to find c.