Add Maths 2012 May-June Paper 21 23 Q12

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The equation of a curve is \(\quad y=2 x^{2}-20 x+37\).

1. Express \(y\) in the form \(a(x+b)^{2}+c,\) where \(a, b\) and \(c\) are integers.
   
   (Answer)

   \(\begin{aligned} y &=2 x^{2}-20 x+37 \\ &=2\left(x^{2}-10 x\right)+37 \\ &=2\left(x^{2}-5 x-5 x+5^{2}-5^{2}\right)+37 \\ &\left.=2\left[(x-5)^{2}-25\right]+37\right] \\ &\left.=2(x-5)^{2}-50+37\right] \\ &=2(x-5)^{2}-13 \end{aligned}\)
2. Write down the coordinates of the stationary point on the curve.
   
   (Answer)

   Coordinate of the stationary point= (5,-13)
   

A function \(f\) is defined by \(f : x \mapsto 2 x^{2}-20 x+37\) for \(x>k\). Given that the function \(f ^{-1}(x)\) exists,

1. write down the least possible value of \(k\),
2. sketch the graphs of \(y= f (x)\) and \(y= f ^{-1}(x)\) on the axes provided,
3. obtain an expression for \(f ^{-1}\).