Question

Describe the transformation.

`y = x {}^(2) \: to \: y = 2(x - 3) {}^(2) + 4`
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Answer 1

A translation of 3 units to the right, followed by a vertical stretch by a factor of 2, followed by a translation of 4 units up.

Explanation:

Transformations

`f(x-a) \implies f(x) \: \textsf{translated $a$ units right}.`

`\begin{aligned} y =a\:f(x) \implies & \textsf{$f(x)$ stretched/compressed vertically by a factor of $a$}.\\& \textsf{If $a > 1$ it is stretched by a factor of $a$}.\\& \textsf{If $0 < a < 1$ it is compressed by a factor of $a$}.\end{aligned}`

`f(x)+a \implies f(x) \: \textsf{translated $a$ units up}`

Therefore, the series of transformations of:

`y=x^2 \quad \textsf{to} \quad y=2(x-3)^2+4\quad \textsf{is}:`

Translated 3 units to the right:

`f(x-3)\implies y=(x-3)^2`

Stretched vertically by a factor of 2:

`2f(x-3)\implies y=2(x-3)^2`

Translated 4 units up:

`2f(x-3)+4\implies y=2(x-3)^2+4`

Therefore, the series of transformations is:

A translation of 3 units to the right, followed by a vertical stretch by a factor of 2, followed by a translation of 4 units up.