# Given f(x) = root3x, what is the translation of g(x) = 5root3(x+2) - 2?

Apr 1, 2018

$g \left(x\right) = 5 f \left(x - 2\right) - 2$

#### Explanation:

First we know that the translation form of any funciton looks like

${x}^{2} \Rightarrow A {\left[B \left(x - C\right)\right]}^{2} + D$
$\sin \left(x\right) \Rightarrow A \sin \left[B \left(x - C\right)\right] + D$
$\ln \left(x\right) \Rightarrow A \ln \left[B \left(x - C\right)\right] + D$
$\sqrt{x} \Rightarrow A \sqrt{B \left(x - C\right)} + D$

With
A~ Vertical stretch, streches the y values by A
B~ Horizontal stretch, streches the x values by $\frac{1}{B}$
C~ Horizontal translation, moves x values over by C
D~ Vertical translation, moves y values up by C

So 5 is the vertical stretch
So -2 is the horizontal translation
So -2 is the vertical translation

So you know the function has been moved 2 to the left and 2 down.

$g \left(x\right) = 5 f \left(x - 2\right) - 2$