Question

find the equation of the square root function described contextually

• its domain is ( x geq -4 ) and its range is ( y leq 8 ).
• it was stretched vertically by a factor of 9.
• ( b = 1 )

use the general formula ( f(x) = asqrt{b(x - h)} + k ).
the corresponding square root function is:
( f(x) = squaresqrt{square(square)}square )

Explanation:

Step1: Identify h from domain

The domain of $f(x)=a\sqrt{b(x-h)}+k$ is $x\geq h$. Given domain $x\geq -4$, so $h=-4$. Substitute into the radicand: $b(x-h)=1\cdot(x-(-4))=x+4$.

Step2: Identify a from vertical stretch

Vertical stretch factor is $|a|$. Given stretch factor 9, and range $y\leq8$ (function opens downward), so $a=-9$.

Step3: Identify k from range

The range of $f(x)=a\sqrt{b(x-h)}+k$ (when $a<0$) is $y\leq k$. Given range $y\leq8$, so $k=8$.

Step4: Assemble the function

Substitute $a=-9$, $b=1$, $h=-4$, $k=8$ into the general formula.
$f(x) = -9\sqrt{1\cdot(x+4)} + 8$

Answer:

$f(x) = -9\sqrt{x+4} + 8$