Question

describe the transformation.
$y = x {}^{2} \: to \: y = 2(x - 3) {}^{2} + 4$
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Answer 1

Answer:

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.

Step-by-step 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.