We had [math]p\left(x\right)=\frac{1}{3\left(x-2\right)}[/math] from the previous part and eventually we want to arrive at [math]g\left(x\right)=\frac{2}{3\left(x-2\right)}-2=2p\left(x\right)-2[/math]

1.[math]q\left(x\right)=p\left(x\right)-2=\frac{1}{3\left(x-2\right)}-2\Longrightarrow[/math] Vertical Translation of 2 units downwards[br]2. [math]g\left(x\right)=2q\left(x\right)=\frac{2}{3\left(x-2\right)}-4\Longrightarrow[/math]Vertical stretching of scale factor 2 vertically

1. [math]q\left(x\right)=2p\left(x\right)=\frac{2}{3\left(x-2\right)}\Longrightarrow[/math]Vertical Stretching of scale factor 2 vertically [br]2. [math]g\left(x\right)=q\left(x\right)-2=\frac{2}{3\left(x-2\right)}-2\Longrightarrow[/math]Vertical translation of 2 units downwards.

Finally, can you sketch the final graph of [math]g\left(x\right)?[/math] You can check your answer using the applet which uses [math]f\left(x\right)=af\left(b\left(x+c\right)\right)+d[/math] below.

Alternatively, you can try using the formula [math]f\left(x\right)=af\left(bx+c\right)+d[/math].[br]Note: the order of transformation is different from the above when using [math]f\left(x\right)=af\left(b\left(x+c\right)\right)+d.[/math]