Nonlinear graphs
Objective. Recognise quadratic and reciprocal graph shapes.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Nonlinear graphs show relationships that are not straight lines, and recognising their shapes helps identify the type of equation that produced them. A quadratic graph, from an equation like y = x², produces a symmetrical U-shaped curve called a parabola, with a single turning point. A reciprocal graph, from an equation like y = k/x, produces two separate curved branches that approach but never touch the axes, called asymptotes. Spotting these shapes from a sketch or table of values allows pupils to match a graph to its equation without plotting every point.
Worked example
Which of these equations produces a U-shaped parabola graph? Choose the correct equation.
- y = 2x + 1
- y = x² + 3
- y = 5/x
- y = 4
- 1A parabola comes from a quadratic equation containing an x² term.
- 2Check each option for an x² term as the defining feature.
- 3y = x² + 3 contains an x² term, matching the parabola shape.
- 4The other options, such as y = 2x + 1 or y = 5/x, are linear or reciprocal, not quadratic.
Answer: y = x² + 3
Concept mastery
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Practice
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Start practisingCommon mistake. Pupils sometimes draw a reciprocal graph as a single continuous curve crossing through the origin, instead of two separate branches that never touch either axis. Remember that reciprocal graphs have two distinct branches and never pass through zero.
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