GCSE Maths · Algebra

Graphs

Straight-line graphs, gradient, and y = mx + c.

202 GCSE-style practice questions

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Graphs, explained point by point

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  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.

  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).

  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.

  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.

  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 −12; perpendicular to −34 is 43. Then find c by substituting the given point, exactly as for parallel lines.

  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.

  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 1x form two separate curves that never touch the axes. Match the shape to the equation family first, then plot key points.

Practice

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A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.

A straight line has gradient 2 and passes through (0, −3). What is its equation?

  1. y = 2x − 3
  2. y = 2x + 3
  3. y = 3x − 2
  4. y = −3x + 2
Show the answer and the working

Answer: y = 2x − 3

With m = 2 and y-intercept −3, the equation y = mx + c becomes y = 2x − 3.

  1. The general equation is y = mx + c.
  2. The gradient is given: m = 2.
  3. The line passes through (0, −3), which is on the y-axis, so c = −3.
  4. Equation: y = 2x − 3.

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