Question

the functions f and g are graphed in the same rectangular coordinate system, shown to the right. if g is obtained from f through a sequence of transformations, find an equation for g. $f(x)=\sqrt{x}$

Explanation:

Step1: Identify parent function

The parent function is $f(x)=\sqrt{x}$.

Step2: Analyze horizontal shift

The graph of $g(x)$ is shifted left 2 units. For a left shift of $h$ units, replace $x$ with $x+h$, so we get $\sqrt{x+2}$.

Step3: Analyze vertical shift

The graph of $g(x)$ is shifted down 3 units. For a down shift of $k$ units, subtract $k$ from the function, so we get $\sqrt{x+2}-3$.

Step4: Verify with points

For $f(x)=\sqrt{x}$, when $x=0$, $f(0)=0$. For $g(x)$, the corresponding point is $(-2,-3)$, which matches $\sqrt{-2+2}-3=-3$. When $x=4$, $f(4)=2$; for $g(x)$, the point $(2,-1)$ gives $\sqrt{2+2}-3=2-3=-1$, which is correct.

Answer:

$g(x)=\sqrt{x+2}-3$