Algebra - Graphs: recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y = 1/x with x ≠ 0 (Foundation tier)

The idea

Recognise and sketch the key graph families. Linear functions (y = mx + c) produce straight lines; interpret 'm' as the gradient (steepness) and 'c' as the y-intercept. Quadratic functions (e.g., y = ax² + bx + c) create a symmetrical U-shaped curve called a parabola. If 'a' is positive, it opens upwards; if 'a' is negative, it opens downwards. Simple cubic functions like y = x³ rise, flatten at the origin, then rise again. More complex cubics can have two turning points and up to three x-intercepts; their shape is rise-fall-rise for a positive x³ term and fall-rise-fall for a negative one. The reciprocal function y = 1/x has two separate branches in opposite quadrants and never touches the axes (asymptotes at x=0, y=0). Note the tiering: the quoted statement is tagged Foundation tier, so the content is examinable at Foundation tier.

What earns the marks

• Generating a correct table of values for the given function.
• Plotting calculated points accurately on the coordinate grid.
• Drawing a single, smooth curve of the correct characteristic shape through the points.
• Correctly identifying the type of function from its graph (e.g., linear, quadratic, cubic).
• Reading a value from a graph, or stating the coordinates of the y-intercept.
• Recognising that the reciprocal graph y = 1/x has two separate branches and asymptotes at x = 0 and y = 0.

Common mistakes and how to put them right

• Joining plotted points with straight line segments instead of a smooth curve.

  Put it right: Draw a single continuous curve that passes through the points.

• Drawing a reciprocal graph (y = 1/x) that crosses the y-axis.

  Put it right: The graph has two separate branches and never touches either axis.

• Assuming a cubic graph has the same line symmetry as a quadratic parabola.

  Put it right: Cubic graphs have rotational symmetry about a point, not the reflective symmetry of a parabola.

• Forgetting that the reciprocal function is undefined at x = 0.

  Put it right: State x ≠ 0 and show the two branches approaching but never meeting the axes.

Exam tips

• Memorise the basic shapes. If a plotted point from your table of values breaks the expected pattern, re-check your calculation as it is likely an error.
• For quadratics, check the sign of the x² term. A positive coefficient gives a U-shape ('smile'), a negative coefficient gives an n-shape ('frown').
• When calculating values for a cubic function, be careful with negative numbers: for example, (-2)³ = -8, not 8.