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

dropdown options:
left of the parent
right of the parent
above the parent
below the parent

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

dropdown options:
left of the parent
right of the parent
above the parent
below the parent

Accepted Answer

{ "quality": "clear", "question_count": 4, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Function Translation", "Vertical Translation" ], "new_concepts": [ "Horizontal Translation", "Square Root Transformations" ], "current_concepts": [ "Function Translation", "Vertical Translation", "Horizontal Translation", "Square Root Transformations" ] } ## Analyze horizontal translations inside the radical Using the *Function Translation* knowledge point $$ \begin{aligned} &y = f(x - h) \text{ shifts the parent function } f(x) \text{ horizontally.}\ &\text{For } y = \sqrt{x + 4} = \sqrt{x - (-4)} \implies h = -4 \text{ (shift left by 4 units).}\ &\text{For } y = \sqrt{x - 4} \implies h = 4 \text{ (shift right by 4 units).} \end{aligned} $$ ## Analyze vertical translations outside the radical Using the *Vertical Translation* knowledge point $$ \begin{aligned} &y = f(x) + k \text{ shifts the parent function } f(x) \text{ vertically.}\ &\text{For } y = \sqrt{x} - 2 \implies k = -2 \text{ (shift down by 2 units).}\ &\text{For } y = \sqrt{x} + 2 \implies k = 2 \text{ (shift up by 2 units).} \end{aligned} $$ ## Match functions to relative positions Using the *Function Translation* and *Vertical Translation* knowledge points $$ \begin{aligned} &y = \sqrt{x + 4} \implies \text{left of the parent}\ &y = \sqrt{x - 4} \implies \text{right of the parent}\ &y = \sqrt{x} - 2 \implies \text{below the parent}\ &y = \sqrt{x} + 2 \implies \text{above the parent} \end{aligned} $$ | Function | Position Relative to Parent | |---|---| | $y = \sqrt{x + 4}$ | left of the parent | | $y = \sqrt{x - 4}$ | right of the parent | | $y = \sqrt{x} - 2$ | below the parent | | $y = \sqrt{x} + 2$ | above the parent | { "elements": [ { "type": "functiongraph", "params": [{"js": "Math.sqrt(x)", "latex": "\sqrt{x}"}, 0, 10], "properties": { "strokeColor": "#583C87", "strokeWidth": 2, "dash": 2, "name": "y = \sqrt{x}", "withLabel": true, "labelPosition": [6, 2.2] } }, { "type": "functiongraph", "params": [{"js": "Math.sqrt(x + 4)", "latex": "\sqrt{x + 4}"}, -4, 10], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = \sqrt{x+4}", "withLabel": true, "labelPosition": [2, 2.8] } }, { "type": "functiongraph", "params": [{"js": "Math.sqrt(x - 4)", "latex": "\sqrt{x - 4}"}, 4, 10], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "name": "y = \sqrt{x-4}", "withLabel": true, "labelPosition": [7, 1.3] } }, { "type": "functiongraph", "params": [{"js": "Math.sqrt(x) - 2", "latex": "\sqrt{x} - 2"}, 0, 10], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "name": "y = \sqrt{x}-2", "withLabel": true, "labelPosition": [6, 0.2] } }, { "type": "functiongraph", "params": [{"js": "Math.sqrt(x) + 2", "latex": "\sqrt{x} + 2"}, 0, 10], "properties": { "strokeColor": "#F28A55", "strokeWidth": 2, "name": "y = \sqrt{x}+2", "withLabel": true, "labelPosition": [4, 4.2] } } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Function Translation" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze horizontal translations inside the radical

Analyze vertical translations outside the radical

Match functions to relative positions

Answer:

Function	Position Relative to Parent
\(y = \sqrt{x - 4}\)	right of the parent
\(y = \sqrt{x} - 2\)	below the parent
\(y = \sqrt{x} + 2\)	above the parent