Question

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}\\) | dropdown
\\(y = \sqrt{x-4}\\) | dropdown
\\(y = \sqrt{x}-2\\) | dropdown
\\(y = \sqrt{x}+2\\) | dropdown

Explanation:

Identify parent function and transformation rules

Using the Horizontal Translation knowledge point
The parent function is \(y = \sqrt{x}\).
For a function \(y = a\sqrt{x - h} + k\):

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

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

Using the Horizontal Translation knowledge point

$$ y = \sqrt{x - (-4)} \implies h = -4 $$

This represents a translation of 4 units to the left.

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

Using the Horizontal Translation knowledge point

$$ y = \sqrt{x - 4} \implies h = 4 $$

This represents a translation of 4 units to the right.

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

We identify the vertical shift parameter \(k\):

$$ y = \sqrt{x} - 2 \implies k = -2 $$

This represents a translation of 2 units down.

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

We identify the vertical shift parameter \(k\):

$$ y = \sqrt{x} + 2 \implies k = 2 $$

This represents a translation of 2 units up.

Answer:

Question 1

For the function \(y = \sqrt{x + 4}\), the position relative to the parent function is:
4 units left

Question 2

For the function \(y = \sqrt{x - 4}\), the position relative to the parent function is:
4 units right

Question 3

For the function \(y = \sqrt{x} - 2\), the position relative to the parent function is:
2 units down

Question 4

For the function \(y = \sqrt{x} + 2\), the position relative to the parent function is:
2 units up