lesson
Roots are where the graph meets the axis
1 · Learn
A root of a function is an input making its output zero. For y = (x - 2)(x + 3), the roots are 2 and -3, so the graph crosses the x-axis there. The y-intercept comes from x = 0.
Do not confuse an x-intercept with the y-intercept or a turning point. Different features answer different questions about the same graph.
2 · Worked example
For y = (x - 2)(x + 3), roots are 2 and -3 and the y-intercept is -6.
3 · Your turn
Find the roots and y-intercept of y = (x - 4)(x + 1).
Check your answer
Roots 4 and -1; y-intercept -4.
4 · Apply your learning
Sketch a suitable quadratic shape using the intercepts and check selected additional values.
Adult guidance and safety
Check prerequisites, units and reasoning. Provide accurate diagrams and varied practice. Extension content needs suitable prior understanding; no GCSE board, tier or examination readiness is implied by this proposed sequence.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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