the cube root function, $f(x) = \sqrt3{x}$, to graph the function $g(x) = \sqrt3{x+6}$.

choose the correct graph below.

Question

the cube root function, $f(x) = \sqrt3{x}$, to graph the function $g(x) = \sqrt3{x+6}$.

choose the correct graph below.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Cube Root Transformations" ], "current_concepts": [ "Cube Root Functions", "Function Transformations", "Cube Root Transformations" ] } ## Identify the horizontal shift of the transformation $$ \begin{aligned} f(x) &= \sqrt[3]{x} \ g(x) &= \sqrt[3]{x+6} = f(x+6) \end{aligned} $$ The transformation $f(x+h)$ with $h = 6$ shifts the graph of $f(x)$ horizontally to the left by $6$ units. ## Determine the key point (inflection point) of the transformed graph $$ \begin{aligned} \text{Inflection point of } f(x) &: (0, 0) \ \text{Inflection point of } g(x) &: (0 - 6, 0) = (-6, 0) \end{aligned} $$ ## Match the inflection point with the given options - Option A: Inflection point is at $(6, 0)$ (shifted right by 6). - Option B: Inflection point is at $(0, 6)$ (shifted up by 6). - Option C: Inflection point is at $(0, -6)$ (shifted down by 6). - Option D: Inflection point is at $(-6, 0)$ (shifted left by 6).

(A) Graph shifted right by 6 units with inflection point at (6, 0)

(B) Graph shifted up by 6 units with inflection point at (0, 6)

(C) Graph shifted down by 6 units with inflection point at (0, -6)

(D) Graph shifted left by 6 units with inflection point at (-6, 0)

{ "elements": [ { "type": "functiongraph", "params": [ { "js": "x >= 0 ? Math.pow(x, 1/3) : -Math.pow(-x, 1/3)", "latex": "\sqrt[3]{x}" }, -10, 10 ], "properties": { "strokeColor": "#583C87", "strokeWidth": 1.5, "dash": 2, "name": "f(x) = \sqrt[3]{x}", "withLabel": true } }, { "type": "functiongraph", "params": [ { "js": "(x+6) >= 0 ? Math.pow(x+6, 1/3) : -Math.pow(-(x+6), 1/3)", "latex": "\sqrt[3]{x+6}" }, -10, 10 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "g(x) = \sqrt[3]{x+6}", "withLabel": true } }, { "type": "point", "params": [ [-6, 0] ], "properties": { "name": "(-6,0)", "size": 4, "strokeColor": "#F2557F", "fillColor": "#F2557F", "withLabel": true } } ], "timestamps": [0.5, 1.5] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Cube Root Transformations" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the horizontal shift of the transformation

Determine the key point (inflection point) of the transformed graph

Match the inflection point with the given options

Answer: