Question

review the graph of ( f(x) = sqrt3{x} ) and the graph of the transformed function ( g(x) ).
if ( g(x) = a cdot f(x + b) ), how is ( f(x) ) transformed to get ( g(x) )?
( \boldsymbol{-2f(x + 4)} )
( -2f(x - 4) )
( -f(x + 4) )
( -f(x - 4) )

Explanation:

Step1: Identify reflection

The graph of $g(x)$ is a vertical reflection of $f(x)$, so $a$ is negative.

Step2: Identify vertical stretch

Compare vertical distances: for $f(0)=0$, $g(-4)=0$; for $f(1)=1$, $g(-3)=-2$. The vertical scale factor is $-2$, so $a=-2$.

Step3: Identify horizontal shift

The inflection point of $f(x)$ is at $(0,0)$; for $g(x)$, it is at $(-4,0)$. This is a shift left by 4 units, so $b=4$.

Step4: Combine transformations

Substitute $a=-2$ and $b=4$ into $g(x)=a\cdot f(x+b)$.
<Expression>
$g(x) = -2f(x+4)$
</Expression>

Answer:

-2f(x + 4)