Graphs of the form \(y=\pm \left( x-h \right)\left( x-k \right)\)

Minimum Curve

For the equation \(y=\left( x-h \right)\left( x-k \right)\),

• the graph opens upwards,

• the graph cuts the \(x\)-axis at \(\left( h,0 \right)\) and \(\left( k,0 \right)\) and

• the graph is symmetrical about the vertical line that passes through the minimum point.

Maximum Curve

For the equation \(y=-\left( x-h \right)\left( x-k \right)\),

• the graph opens downwards,

• the graph cuts the \(x\)-axis at \(\left( h,0 \right)\) and \(\left( k,0 \right)\) and

• the graph is symmetrical about the vertical line that passes through the maximum point.

Sketching of graphs of quadratic functions given in the form of

\(y=\pm \left( x-h \right)\left( x-k \right),h>0,k>0\)

Step 1: State the coefficient of \({{x}^{2}}\) to determine if the graph opens upwards or downwards.

• If the coefficient of \({{x}^{2}}<0\), the graph open downwards (maximum curve)

• If the coefficient of \({{x}^{2}}>0\), the graph open upwards (minimum curve)

Step 2: Obtain the \(x\)-intercepts by substituting \(y=0\) into the equation.

• Cuts the \(x\)-axis, \(\left( h,0 \right)\) and \(\left( k,0 \right)\)

Step 3: Obtain the y-intercept by substituting \(x=0\) into the equation.

• Cuts the \(y\)-axis, \(\left( 0,y \right)\)

Step 4: Sketch the graph.