Recognising power graphs
The family \(y=ax^n\), with non-zero constant \(a\), includes \(n=-2,-1,0,1,2,3\). These sketches use \(a=1\); for \(n=0\), the constant function is written \(y=1\), including at \(x=0\).
\(y=1\): constant, \(n=0\).
\(y=x\): straight line, \(n=1\).
\(y=x^2\): square, \(n=2\).
\(y=x^3\): cube, \(n=3\).
\(y=x^{-1}=\dfrac1x\), \(x\ne0\).
\(y=x^{-2}=\dfrac1{x^2}\), \(x\ne0\).
The reciprocal graphs have separate branches: never join across \(x=0\). For \(a>0\), the reciprocal changes sign with \(x\), while the reciprocal-square stays positive. A negative coefficient reverses the signs of the ordinates.
Sums such as \(y=x^3-x+1\) need a table or further information; their shapes are not determined by one power alone.
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