Graphs

1. Plot straight-line graphs from a table of values

   Substitute a few x values into the equation to build a table, plot the points, and join them with a ruler. For y = 2x − 1: x = 0, 1, 2 gives y = −1, 1, 3. If one point does not line up with the others, recheck that substitution.

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2. Find the gradient and y-intercept of a line, and read them from y = mx + c

   In y = mx + c, m is the gradient (the rise for every 1 across) and c is where the line crosses the y-axis. From a graph, gradient = change in y ÷ change in x between two clear points. y = 3x − 2 climbs 3 for every 1 across and crosses at (0, −2).

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3. Write down the equation of a straight line from its graph

   Read off the y-intercept c, measure the gradient m from two points on the line, and assemble y = mx + c. A line through (0, 4) that drops 2 for every 1 across is y = −2x + 4. Downhill lines always have negative gradients.

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4. Find equations of lines parallel to a given line

   Parallel lines share the same gradient and differ only in c. A line parallel to y = 3x + 1 through (0, −5) is y = 3x − 5. If the point is not on the y-axis, substitute it into y = 3x + c and solve for c.

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5. Find equations of perpendicular linesHigher only

   Perpendicular gradients multiply to −1 - flip the fraction and change the sign. Perpendicular to gradient 2 is gradient $\frac{-1}{2}$; perpendicular to $\frac{-3}{4}$ is $\frac{4}{3}$. Then find c by substituting the given point, exactly as for parallel lines.

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6. Draw and interpret real-life graphs - distance–time and conversion graphs

   On a distance–time graph, the gradient is the speed: steeper means faster and a flat section means stopped. Conversion graphs turn one unit into another - read across and down. Always check the axis scales before reading anything off.

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7. Recognise and sketch quadratic, cubic and reciprocal graphs

   Quadratics (x²) make a symmetrical U - or an upside-down U when the x² term is negative. Cubics (x³) sweep from one corner of the grid to the other with a wiggle; reciprocals $\frac{1}{x}$ form two separate curves that never touch the axes. Match the shape to the equation family first, then plot key points.

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• Working out the gradient as change in x ÷ change in y, which turns it upside down.

  Gradient is change in y ÷ change in x. From (1, 2) to (3, 8) that is (8 − 2) ÷ (3 − 1) = 6 ÷ 2 = 3.

• Giving a positive gradient to a line that slopes downhill from left to right.

  A line going down from left to right has a negative gradient. If y drops by 2 for every 1 across, the gradient is −2.

• Reading the gradient of 2y = 6x + 4 as 6, before rearranging to y = mx + c.

  Get y on its own first. Dividing every term by 2 gives y = 3x + 2, so the gradient is 3 and the line crosses the y-axis at 2.

• Counting squares to find a gradient without checking the scales on the axes.

  Use the numbers on the axes, not the grid squares: if y goes from 10 to 40 while x goes from 0 to 5, the gradient is 30 ÷ 5 = 6.

• Doing only half the job for a perpendicular gradient, so perpendicular to 2 comes out as −2 or $\frac{1}{2}$.Higher only

  Flip the fraction and change the sign, so perpendicular to 2 is $\frac{-1}{2}$. Check: the two gradients multiply to −1, and 2 × $\frac{-1}{2}$ = −1.

More Algebra topics

• Linear Equations
• Expanding Brackets
• Quadratic Equations
• Sequences
• Simultaneous Equations
• Inequalities

Keep reading

• How do I revise for GCSE maths?
• What's on the GCSE maths syllabus?
• What are the GCSE Maths grade boundaries?
• The GCSE Maths syllabus, explained