Question

exploring translations of square root functions
move the slider on the graph on the right to graph
each function and describe the translation.
function position relative to parent
$y = \sqrt{x + 4}$
$y = \sqrt{x - 4}$
$y = \sqrt{x} - 2$
$y = \sqrt{x} + 2$
done
$y = \sqrt{x - h} + k$ $h = -1$ $k = 1$
$y = \sqrt{x}$

Explanation:

To determine the translation of each square root function relative to the parent function \( y = \sqrt{x} \), we use the transformation form \( y = \sqrt{x - h} + k \), where:

• \( h \) represents the horizontal shift (if \( h > 0 \), shift right; if \( h < 0 \), shift left).
• \( k \) represents the vertical shift (if \( k > 0 \), shift up; if \( k < 0 \), shift down).

Step 1: Analyze \( y = \sqrt{x + 4} \)

Rewrite \( y = \sqrt{x + 4} \) as \( y = \sqrt{x - (-4)} + 0 \).
Here, \( h = -4 \) and \( k = 0 \).
Since \( h = -4 < 0 \), the graph shifts 4 units to the left.

Step 2: Analyze \( y = \sqrt{x - 4} \)

Rewrite \( y = \sqrt{x - 4} \) as \( y = \sqrt{x - 4} + 0 \).
Here, \( h = 4 \) and \( k = 0 \).
Since \( h = 4 > 0 \), the graph shifts 4 units to the right.

Step 3: Analyze \( y = \sqrt{x} - 2 \)

Rewrite \( y = \sqrt{x} - 2 \) as \( y = \sqrt{x - 0} + (-2) \).
Here, \( h = 0 \) and \( k = -2 \).
Since \( k = -2 < 0 \), the graph shifts 2 units down.

Step 4: Analyze \( y = \sqrt{x} + 2 \)

Rewrite \( y = \sqrt{x} + 2 \) as \( y = \sqrt{x - 0} + 2 \).
Here, \( h = 0 \) and \( k = 2 \).
Since \( k = 2 > 0 \), the graph shifts 2 units up.

Answer:

• \( y = \sqrt{x + 4} \): 4 units left
• \( y = \sqrt{x - 4} \): 4 units right
• \( y = \sqrt{x} - 2 \): 2 units down
• \( y = \sqrt{x} + 2 \): 2 units up