which of the following graphs could be the graph of the function $f(x) = x^3 + 8$?

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Cubic Function Analysis", "Polynomial Roots", "Vertical Translation" ], "new_concepts": [], "current_concepts": [ "Cubic Function Analysis", "Polynomial Roots", "Vertical Translation", "Function Graphing" ] } ## Analyze key features of the function Using the *Cubic Function Analysis* and *Vertical Translation* knowledge points $$ \begin{aligned} &f(x) = x^3 + 8\ &\text{y-intercept: } f(0) = 0^3 + 8 = 8\ &\text{x-intercept: } x^3 + 8 = 0 \implies x^3 = -8 \implies x = -2 \end{aligned} $$ ## Match features to the given graphs Using the *Polynomial Roots* knowledge point $$ \begin{aligned} &\text{Key Point 1: } (0, 8)\ &\text{Key Point 2: } (-2, 0)\ &\text{Graph 1 (top): Passes through } (-2, 0) \text{ and } (0, 8)\ &\text{Graph 2 (bottom): Parabola with vertex below the x-axis, representing a quadratic function} \end{aligned} $$

(A) A cubic curve passing through the x-axis at $(-2, 0)$ and the y-axis at $(0, 8)$

(B) A parabola with its vertex below the x-axis

{ "elements": [ { "type": "functiongraph", "params": [ { "js": "Math.pow(x, 3) + 8", "latex": "x^3 + 8" }, -3.5, 1.5 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "f(x) = x^3 + 8", "withLabel": true } }, { "type": "point", "params": [ [-2, 0] ], "properties": { "name": "(-2, 0)", "color": "#F2557F", "size": 4, "withLabel": true } }, { "type": "point", "params": [ [0, 8] ], "properties": { "name": "(0, 8)", "color": "#F2557F", "size": 4, "withLabel": true } } ], "timestamps": [ 0.5, 1.0 ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Cubic Function Analysis" ] }

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Explanation:

Analyze key features of the function

Match features to the given graphs

Answer: