Graphs of Power Functions \(y=a{{x}^{n}}\), where \(n=3,2,1,0,-1\) and \(-2\)

For \(n=0\),\(y=a\)

• For \(x\ne0\), \(x^0=1\). The constant function \(y=a\) shown here is defined at every real input, including \(0\).

For \(n=1\), \(y=ax\)

• It is a straight line graph, either sloping upwards or sloping downwards depending on its gradient which is the value of \(a\).

For \(n=2\), \(y=a{{x}^{2}}\)

• It is a quadratic graph, either opening upwards or downwards depending on the value of \(a\).

For \(n=3\), \(y=a{{x}^{3}}\)

• It is a cubic graph.

For \({n = -1}\),
\({y = ax^{-1}}\)
\({= \frac{a}{x}}\)

• It is a reciprocal graph.

• The graph occurs in two separate parts but it is considered as a single graph.

• When \(x=0\), the function \(y=\frac{a}{x}\) is not defined. This means that there is a break at \(x=0\).

• The graph approach very close to both the \(x\)-axis and \(y\)-axis but never meet them. The axes are known as asymptotes.

For \({n = -2}\),
\({y = ax^{-2}}\)
\({= \frac{a}{x^2}}\)

• It is a reciprocal graph.

• The graph occurs in two separate parts but it is considered as a single graph.

• When \(x=0\), the function \(y=\frac{a}{{{x}^{2}}}\) is not defined. This means that there is a break at \(x=0\).

• The graph approach very close to both the \(x\)-axis and \(y\)-axis but never meet them. The axes are known as asymptotes.

• Since \({{x}^{2}}>0\), hence

  when \(a>0\), the graph lies above the horizontal axis and

  when \(a<0\), the graph lies below the horizontal axis