Graphs: recognise, sketch and interpret graphs of linear functions and quadratic functions

The idea

Linear functions have the form y = mx + c and graph as straight lines: m is the gradient and c is the y-intercept. Quadratic functions have the form y = ax² + bx + c and graph as parabolas; a > 0 opens upwards, a < 0 opens downwards, and the y-intercept is c. Recognising means matching an equation to its graph shape and key features. Sketching means drawing a clear curve or line with labelled intercepts, turning point and axis of symmetry where relevant, without needing exact plotting. Interpreting means reading information from a given graph, such as gradient, intercepts, roots and the turning point.

For example: y = 2x − 3 is a line crossing the y-axis at −3 with gradient 2, while y = x² − 4 is a parabola crossing the x-axis at x = −2 and x = 2 with a minimum at (0, −4).

What earns the marks

• Identify a linear function from its equation or graph and state its gradient and y-intercept.
• Recognise a quadratic function from its equation or graph and identify the direction of opening from the sign of the coefficient of x².
• Sketch a straight-line graph using the gradient and y-intercept, or by plotting two convenient points.
• Sketch a quadratic graph showing the correct shape, y-intercept, roots where they exist, and turning point.
• Interpret key features from a graph, including intercepts, gradient, roots and the coordinates of the turning point.

Common mistakes and how to put them right

• Confusing the gradient and y-intercept in y = mx + c.

  Put it right: The coefficient of x is the gradient and the constant is the y-intercept.

• Drawing a quadratic as a straight line or with the wrong curvature.

  Put it right: A quadratic must be a smooth parabola, not a V-shape or a line.

• Assuming a parabola always crosses the x-axis.

  Put it right: It may have no real roots, one repeated root or two roots.

• Reading the turning point as the y-intercept.

  Put it right: The turning point is the maximum or minimum, while the y-intercept occurs at x = 0.

Exam tips

• Label the y-intercept and any roots on your sketch to show the key features clearly.
• Use the sign of the coefficient of x² to decide whether the parabola opens upwards or downwards before drawing it.
• For a linear graph, plot at least two points or use the gradient and y-intercept to ensure the line is straight and correctly positioned.
• When interpreting a graph, read coordinates from the axes carefully and state units if the context provides them.