lesson
A cubic changes sign through its central root
1 · Learn
The basic cubic y=x³ has an S-shaped graph through the origin. Negative inputs give negative outputs and positive inputs give positive outputs; y(-x)=-y(x), so the graph has rotational symmetry about the origin.
Multiplying by a positive constant makes output magnitudes larger without changing the root. A negative coefficient reflects the graph in the x-axis. Plot a table on both sides of zero and join it with a smooth curve rather than straight segments pretending the gradient is constant.
A cubic is different from the reciprocal y=1/x. The cubic is defined at zero and crosses there; the reciprocal is undefined at zero and has separate branches approaching the axes.
2 · Worked example
For y=2x³, x=-2,-1,0,1,2 gives y=-16,-2,0,2,16. The graph crosses at (0,0), and its signs reverse when x changes sign.
3 · Your turn
Make a five-point table for y=-x³ using x=-2,-1,0,1,2. State the root and describe the reflection from y=x³.
Check your answer
Outputs 8,1,0,-1,-8; root x=0. The negative coefficient reflects y=x³ in the x-axis.
4 · Apply your learning
Plot y=x³ and y=1/x on separate axes. Annotate the value or restriction at x=0 and compare their shapes.
Adult guidance and safety
This is England national-curriculum core teaching, not an exam-board specification. Check prerequisite understanding, require clear working and use additional varied practice before judging fluency.
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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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