Question

select the correct answer.
the parent square root function, f, is transformed to create function g.
$g(x) = \sqrt{x + 3} - 4$
which statement is true?
a. the graph of f is translated 3 units to the right and 4 units up.
b. the graph of f is translated 3 units to the left and 4 units down.
c. the graph of f is translated 4 units to the right and 3 units down.
d. the graph of f is translated 4 units to the left and 3 units up.

Explanation:

Step1: Recall transformation rules

For a function \( y = \sqrt{x - h} + k \), the horizontal translation is \( h \) units (right if \( h>0 \), left if \( h<0 \)) and vertical translation is \( k \) units (up if \( k>0 \), down if \( k<0 \)). The parent square root function is \( f(x)=\sqrt{x} \). The given function is \( g(x)=\sqrt{x + 3}-4=\sqrt{x-(-3)}+(-4) \).

Step2: Analyze horizontal translation

Here, \( h=-3 \). Since \( h<0 \), the graph is translated \( |h| = 3 \) units to the left.

Step3: Analyze vertical translation

Here, \( k = - 4 \). Since \( k<0 \), the graph is translated \( |k| = 4 \) units down.

Answer:

B. The graph of \( f \) is translated 3 units to the left and 4 units down.