\(y=a{{x}^{n}}\) \(a>0\) \(a<0\) Notes 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

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

Secondary 3

\(y=a{{x}^{n}}\)	\(a>0\)	\(a<0\)	Notes
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