Question

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Explanation:

Identify parent function and transformations

Using the Square Root Transformations knowledge point
The parent function is \(f(x) = \sqrt{x}\).
A transformed square root function is written as \(y = a\sqrt{x - h} + k\), where:

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

Analyze the first function

Using the Square Root Transformations knowledge point
For the function \(y = \sqrt{x - 3} + 2\):

• The term \(x - 3\) inside the radical indicates a horizontal shift. Since \(h = 3\), the graph is translated 3 units to the right.
• The term \(+ 2\) outside the radical indicates a vertical shift. Since \(k = 2\), the graph is translated 2 units up.

Analyze the second function

Using the Square Root Transformations knowledge point
For the function \(y = \sqrt{x + 3} - 2\):

• The term \(x + 3\) inside the radical indicates a horizontal shift. Since \(h = -3\), the graph is translated 3 units to the left.
• The term \(- 2\) outside the radical indicates a vertical shift. Since \(k = -2\), the graph is translated 2 units down.

Answer:

For the function \(y = \sqrt{x - 3} + 2\):
The graph of \(y = \sqrt{x}\) is translated 3 units right and 2 units up.

For the function \(y = \sqrt{x + 3} - 2\):
The graph of \(y = \sqrt{x}\) is translated 3 units left and 2 units down.