Which transformations give us the graph of $y=2^{x^2-4x+3}$?

Which transformations give us the graph of $y=2^{x^2-4x+3}$?

Question

The graph of the function $y=2^{x^2-4x+3}$ can be obtained from the graph of $y = 2^{x^2}$ by

1. a stretch parallel to the $y$-axis followed by a translation parallel to the $y$-axis.
2. a translation parallel to the $x$-axis followed by a stretch parallel to the $y$-axis.
3. a translation parallel to the $x$-axis followed by one parallel to the $x$-axis.
4. a translation parallel to the $x$-axis followed by a reflection in the $y$-axis.
5. a reflection parallel to the $y$-axis followed by a translation parallel to the $y$-axis.

Suggestion

The starting and finishing functions differ by $x^2 \mapsto x^2-4x+3$. Could we rewrite the latter to look like a transformation?

Solution

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).