2015 RI P1 Q8

1. 1. If \(\mathrm{f}\left( x \right)=\sin 2x\), describe the transformation which would transform the graph of \(y=\mathrm{f}\left( x \right)\) to the graph of \(y=-\mathrm{f}''\left( x \right)\).
   2. If \(\mathrm{g}\left( x \right)={{2}^{kx+3}}\), \(k>0\), describe a sequence of two transformations which would transform the graph of \(y=\mathrm{g}\left( x \right)\) to the graph of \(y=\frac{1}{\mathrm{g}\left( x \right)}\).
2. The diagram shows the graph of \(y=\mathrm{h}\left( x \right)\) with \(\left( -2,4 \right)\) as its maximum point. The curve passes through the points \(\left( 1,0 \right)\), \(\left( 3,0 \right)\) and \(\left( 0,2 \right)\), and the lines \(x=2\) and \(y=2\) are asymptotes of the curve.
   
   The following properties are known about the graph of \(y=a\mathrm{h}\left( x+b \right)\), where \(a\) and \(b\) are real constants.
   1. \(y=a\mathrm{h}\left( x+b \right)\ge -1\) for all values of \(x\) where it is defined.
   2. The two asymptotes of the graph of \(y=a\mathrm{h}\left( x+b \right)\) intersect in the region where \(x<0\).

Write down the range of values of \(a\) and \(b\).