Question

what radical function is represented in the graph?
$f(x)=\square$
(simplify your answer.)

Explanation:

Step1: Identify base radical form

The graph matches the shape of a square root function shifted horizontally and vertically, so we use the form:
$f(x) = a\sqrt{x - h} + k$
where $(h,k)$ is the horizontal/vertical shift, and $a$ is the stretch factor.

Step2: Use point $(-1,3)$ for shift

The function's domain starts at $x=-1$, so $h=-1$. Substitute $h=-1$, $x=-1$, $f(x)=3$:
$3 = a\sqrt{-1 - (-1)} + k$
$3 = a\sqrt{0} + k$
$k=3$

Step3: Use point $(0,4)$ to find $a$

Substitute $h=-1$, $k=3$, $x=0$, $f(x)=4$:
$4 = a\sqrt{0 - (-1)} + 3$
$4 = a\sqrt{1} + 3$
$4 = a + 3$
$a=1$

Step4: Write final function

Substitute $a=1$, $h=-1$, $k=3$ into the base form:
$f(x) = \sqrt{x + 1} + 3$

Answer:

$f(x)=\sqrt{x+1}+3$