Find the minimum point on the curve \(y=x^{2}-4x+2\). Find the equation of other quadratic graphs with the same minimum point. Can you find one that passes through the origin \((0, 0)\)?
Complete the square to give \(y=(x-2)^{2}-2\) so the minimum point is at \((2,-2)\).
Other graphs will have equations of the form \(y=-2+k(x-2)^{2}\) for any value positive value of \(k\).
(Specific equations with integer coefficients include \(y=2^{x}=-8x+6\) and \(y=3x^{2}-12x+10\))
To go through the origin, \(y=0\) when \(x=0\) so \(0=2+k(0-2)^{2}\) giving \(0=-2+4k\) so \(k=\frac1 2\)
This gives the equation \(y=-2+\frac 1 2(x-2)^{2}=\frac 1 2 x(x-4)\)
Alternative Approach – use graph drawing software to sketch \(y=-2+k(x-2)^{2}\) and explore the graph for different values of \(k\).