Cubic and reciprocal graphs

Objective. Recognise, sketch and interpret cubic and reciprocal forms.

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Learn: the key idea

Cubic graphs follow y=ax³+bx²+cx+d and have a characteristic S-shape, while reciprocal graphs follow y=k/x and form two curved branches that never touch the axes. A positive cubic like y=x³ rises from bottom-left to top-right through the origin, flattening briefly at points of inflection. A reciprocal graph like y=1/x has asymptotes at x=0 and y=0, meaning the curve gets arbitrarily close to these lines but never crosses them. To sketch either, plot a few key points, such as where x=1, 2, −1, −2, and note symmetry: y=1/x has rotational symmetry about the origin. Always mark any asymptotes with dashed lines when sketching reciprocal graphs.

Worked example

State the equations of the two asymptotes of the reciprocal graph y = 6/x.

1 $A reciprocal graph y = k/x is undefined when x = 0$
2 $This gives a vertical asymptote at x = 0$
3 $As x becomes very large, y approaches 0 but never reaches it$
4 $This gives a horizontal asymptote at y = 0$

Answer: x = 0 and y = 0

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Common mistake. Pupils draw the reciprocal graph y=k/x passing through the origin, forgetting x=0 is undefined since division by zero is impossible. The curve must approach but never touch either axis. Always leave a gap around x=0 and y=0 on reciprocal sketches.

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