The graph of the function \[y=2^{x^2-4x+3}\] can be obtained from the graph of \(y = 2^{x^2}\) by
- a stretch parallel to the \(y\)-axis followed by a translation parallel to the \(y\)-axis.
- a translation parallel to the \(x\)-axis followed by a stretch parallel to the \(y\)-axis.
- a translation parallel to the \(x\)-axis followed by one parallel to the \(x\)-axis.
- a translation parallel to the \(x\)-axis followed by a reflection in the \(y\)-axis.
- a reflection parallel to the \(y\)-axis followed by a translation parallel to the \(y\)-axis.
Completing the square, we can see \(x^2-4x+3 = (x-2)^2-1\), and thus \(y=2^{x^2-4x+3} = 2^{(x-2)^2-1} = \dfrac{1}{2}\times 2^{(x-2)^2}\).
So if \(f(x) = 2^{x^2}\), then \(y=\dfrac{1}{2}\times 2^{(x-2)^2} = \dfrac{1}{2}f(x-2)\).
Now \(y=af(x+b)\) represents stretching \(y = f(x)\) in the \(y\)-direction by a scale factor of \(a\), and translating left by \(b\).
We can apply these transformations in either order for the same result. So the answer is (b).
