Graphs

OCR

GCSE

Graphs serve as the geometric representation of algebraic relationships, requiring candidates to translate fluently between equations, tables of values, and Cartesian plots. Mastery involves not only accurate plotting of linear, quadratic, and reciprocal functions but also the interpretation of key features such as gradients, intercepts, and turning points in both abstract and real-world contexts. Higher-level analysis demands the ability to model physical phenomena, such as velocity and acceleration, using tangents and areas under curves, linking pure mathematics to kinematic applications.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

Subtopics in this area

Graphs

What You Need to Demonstrate

Key skills and knowledge for this topic

Award B2 for fully correct plotting of points within half a small square tolerance
Award B1 for a smooth curve passing through all plotted points (do not credit straight line segments)
Award M1 for drawing a tangent at the specified x-value to estimate the gradient of a curve
Award B1 for correctly identifying the y-intercept as the constant term in a linear equation
Award M1 for a method to calculate gradient using a triangle drawn on the line (change in y / change in x)
Award B1 for plotting points correctly within a tolerance of ±½ a small square
Award M1 for a correct method to calculate gradient (change in y divided by change in x), clearly shown on the graph or via calculation
Credit responses that draw a smooth, continuous curve for quadratic or cubic functions; reject ruled line segments joining points

Example Examiner Feedback

Real feedback patterns examiners use when marking

"Your plotting is accurate, but you lost a mark for using straight lines on a curve — practice drawing smooth freehand curves."
"You correctly identified the gradient, but check the units — what does this rate of change represent in the context of the question?"
"Remember that 'intercept' refers to where the line crosses the y-axis; ensure you state the coordinate (0, c), not just the value c."
"For the area under the curve, your estimation method needs to be clearer — show the trapezia or shapes you used to calculate the total."

Marking Points

Key points examiners look for in your answers

Award B2 for fully correct plotting of points within half a small square tolerance
Award B1 for a smooth curve passing through all plotted points (do not credit straight line segments)
Award M1 for drawing a tangent at the specified x-value to estimate the gradient of a curve
Award B1 for correctly identifying the y-intercept as the constant term in a linear equation
Award M1 for a method to calculate gradient using a triangle drawn on the line (change in y / change in x)
Award B1 for plotting points correctly within a tolerance of ±½ a small square
Award M1 for a correct method to calculate gradient (change in y divided by change in x), clearly shown on the graph or via calculation
Credit responses that draw a smooth, continuous curve for quadratic or cubic functions; reject ruled line segments joining points
Award 1 mark for correctly identifying the y-intercept as the constant term 'c' when the line equation is in the form y = mx + c
For velocity-time graphs (Higher), award marks for splitting the area under the curve into triangles and rectangles to calculate distance

Examiner Tips

Expert advice for maximising your marks

💡When asked to 'solve' using a graph, draw vertical lines from the x-axis to the curve and horizontal lines to the y-axis to show your method clearly
💡For 'Sketch' questions, you do not need graph paper, but you must label key features: axes, origin, intercepts, and turning points
💡Always calculate the value of one small square on both axes before reading or plotting points to avoid scale errors
💡If calculating a gradient, choose points as far apart as possible on the line to maximise accuracy
💡When asked to 'estimate' a value from a graph, you must draw construction lines (dashed lines) on the grid to show where you read the value; examiners look for this evidence
💡For questions involving y = mx + c, always rearrange the equation into this form first if it is given as ax + by = c; otherwise, the gradient identification will be incorrect
💡In real-life graphs, check the units on the axes carefully; the gradient often represents a rate (e.g., cost per unit, speed) and the area often represents a quantity (e.g., distance)
💡Use a sharp pencil for plotting; thick lines can lead to accuracy penalties if they exceed the tolerance of half a small square

Common Mistakes

Pitfalls to avoid in your exam answers

Joining points on a quadratic or cubic graph with straight line segments (using a ruler) instead of drawing a smooth curve
Misreading the scale on the axes, particularly when 1 large square represents 2, 5, or 0.5 units
Confusing equations of vertical lines (x = c) with horizontal lines (y = c)
Feathering or sketching multiple lines for a curve; the final answer must be a single continuous line
Joining points with straight ruled lines instead of a smooth curve when plotting non-linear functions (quadratics, cubics)
Misinterpreting the scale on the axes, particularly when one large square represents a value other than 1 or 10
Calculating the gradient as 'change in x / change in y' or failing to recognize a negative gradient for downward sloping lines
Confusing the concepts of 'sketch' (showing general shape and key intercepts) with 'plot' (requiring accurate coordinates on a grid)

Study Guide Available

Comprehensive revision notes & examples

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Key Terminology

Essential terms to know

Linear functions and the equation of a straight line (y = mx + c)

Quadratic, cubic, and reciprocal graphs

Real-life graphs: Distance-time and velocity-time

Gradients as rates of change and areas as physical quantities

Graphical solutions to simultaneous equations

Linear functions and coordinate geometry (y = mx + c)

Quadratic, cubic, and reciprocal graphs

Real-life graphs and kinematics (Distance-Time, Velocity-Time)

Gradients as rates of change and areas under curves

Graphical solutions to simultaneous equations

Likely Command Words

How questions on this topic are typically asked

Plot

Draw

Sketch

Interpret

Estimate

Calculate

Find

Practical Links

Related required practicals

•{"code":"Kinematics","title":"Velocity-Time Graphs","relevance":"Linking gradient to acceleration and area to distance traveled"}
•{"code":"Finance","title":"Conversion Graphs","relevance":"Using linear graphs to convert currencies or calculate utility bills"}

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