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{ "quality": "incomplete", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Quadratic Transformations", "Parabola Vertex", "Trajectory Modeling" ], "current_concepts": [ "Quadratic Transformations", "Parabola Vertex", "Trajectory Modeling" ] } ## Identify the missing context and define standard functions $$ \begin{aligned} &f_1(x) = x^2 \quad \text{(Parent Function)}\ &f_2(x) = 2x^2 \quad \text{(Vertical Stretch)}\ &f_3(x) = \frac{1}{2}x^2 \quad \text{(Vertical Compression)}\ &f_4(x) = -x^2 \quad \text{(Reflection across x-axis)}\ &f_5(x) = (x-3)^2 + 2 \quad \text{(Horizontal/Vertical Shift)} \end{aligned} $$ ## Analyze transformation effects on vertex and shape $$ \begin{aligned} &\text{Vertical Stretch } (a > 1): \text{Vertex } (0,0), \text{ narrower shape}\ &\text{Vertical Compression } (0 < a < 1): \text{Vertex } (0,0), \text{ wider shape}\ &\text{Reflection } (a < 0): \text{Vertex } (0,0), \text{ opens downward}\ &\text{Translation } (x-h)^2+k: \text{Vertex shifts to } (h,k), \text{ shape unchanged}\ &\text{Most significant visual change: Reflection } (a < 0) \text{ due to orientation reversal} \end{aligned} $$ ## Apply transformations to trajectory modeling $$ \begin{aligned} &\text{High, steep arc (narrow, tall): } y = -2(x-h)^2 + k \quad (a < -1)\ &\text{Flatter, longer trajectory (wide, low): } y = -0.2(x-h)^2 + k \quad (-1 < a < 0) \end{aligned} $$ ## Question 2 The parent function and its typical transformations are plotted on the coordinate axes below. ## Question 3 * **Vertical Stretch ($a > 1$):** Vertex remains $(0,0)$; the parabola becomes narrower. * **Vertical Compression ($0 < a < 1$):** Vertex remains $(0,0)$; the parabola becomes wider. * **Reflection ($a < 0$):** Vertex remains $(0,0)$; the parabola opens downward. * **Horizontal/Vertical Shifts:** Vertex shifts to $(h,k)$; the shape remains unchanged. * **Most Significant Change:** Reflection across the x-axis, as it completely reverses the orientation of the graph. ## Question 4 * **High, steep arc:** Requires a vertical stretch and downward opening, modeled by a larger absolute value of $a$ (e.g., $y = -2(x-3)^2 + 8$). * **Flatter, longer trajectory:** Requires a vertical compression and downward opening, modeled by a smaller absolute value of $a$ (e.g., $y = -0.2(x-5)^2 + 3$). { "elements": [ { "type": "functiongraph", "params": [{"js": "x*x", "latex": "x^2"}, -5, 5], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = x^2", "withLabel": true } }, { "type": "functiongraph", "params": [{"js": "2*x*x", "latex": "2x^2"}, -5, 5], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "name": "y = 2x^2", "withLabel": true } }, { "type": "functiongraph", "params": [{"js": "0.5*x*x", "latex": "0.5x^2"}, -5, 5], "properties": { "strokeColor": "#55DDF2", "strokeWidth": 2, "name": "y = 0.5x^2", "withLabel": true } }, { "type": "functiongraph", "params": [{"js": "-x*x", "latex": "-x^2"}, -5, 5], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "name": "y = -x^2", "withLabel": true } } ], "timestamps": [0.1, 0.2, 0.3, 0.4] } { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Quadratic Transformations" ] }

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Identify the missing context and define standard functions

Analyze transformation effects on vertex and shape

Apply transformations to trajectory modeling

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