Question

identify all statements that are true when graphing the following square root function:
$f(x) = -sqrt{x - 2} + 3$

i. the horizontal shift is left 2 spaces.
ii. the vertical shift is up 3 spaces.
iii. the square root graph will flip.
iv. the square root graph will stretch by a factor of 2.

\bigcirc a. ii and iii only
\bigcirc b. i, ii, and iii only
\bigcirc c. all statements are true
\bigcirc d. iii and iv only

Explanation:

Step1: Analyze horizontal shift

For $f(x)=-\sqrt{x-2}+3$, compare to parent $g(x)=\sqrt{x}$. The $x-2$ means shift right 2, so Statement I is false.

Step2: Analyze vertical shift

The $+3$ outside the radical shifts the parent graph up 3, so Statement II is true.

Step3: Analyze reflection (flip)

The negative sign in front of the radical reflects the parent graph over the x-axis (flips it), so Statement III is true.

Step4: Analyze vertical stretch

There is no coefficient with absolute value ≠1 on the radical, so no stretch by factor 2. Statement IV is false.

Step5: Identify true statements

Only Statements II and III are true, matching option a.

Answer:

a. II and III only