7. what horizontal translation is applied to $y = x^2$ if the translation image graph passes through the point $(5, 16)$?

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Function Transformations", "Horizontal Translation" ], "new_concepts": [], "current_concepts": [ "Function Transformations", "Horizontal Translation", "Quadratic Functions" ] } ## Formulate the translation equation Using the **Horizontal Translation** and **Function Transformations** knowledge points $$ y = (x - h)^2 $$ ## Substitute the given point Using the **Horizontal Translation** knowledge point $$ \begin{aligned} 16 &= (5 - h)^2 \ 5 - h &= \pm 4 \end{aligned} $$ ## Solve for the translation parameter Using the **Horizontal Translation** knowledge point $$ \begin{aligned} h_1 &= 5 - 4 = 1 \ h_2 &= 5 + 4 = 9 \end{aligned} $$ ## Interpret the geometric translations Using the **Horizontal Translation** knowledge point $$ \begin{aligned} h = 1 &\implies \text{Translation of 1 unit to the right} \ h = 9 &\implies \text{Translation of 9 units to the right} \end{aligned} $$ A horizontal translation of 1 unit to the right, or 9 units to the right. { "elements": [ { "type": "functiongraph", "params": [{"js": "x*x", "latex": "y = x^2"}, -10, 10], "properties": { "strokeColor": "#583C87", "strokeWidth": 1, "dash": 1, "name": "y = x^2", "withLabel": true } }, { "type": "functiongraph", "params": [{"js": "(x-1)*(x-1)", "latex": "y = (x-1)^2"}, -10, 10], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = (x-1)^2", "withLabel": true } }, { "type": "functiongraph", "params": [{"js": "(x-9)*(x-9)", "latex": "y = (x-9)^2"}, -10, 10], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "name": "y = (x-9)^2", "withLabel": true } }, { "type": "point", "params": [[5, 16]], "properties": { "name": "(5, 16)", "size": 4, "color": "#F2557F", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Horizontal Translation" ] }

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Question

Explanation:

Formulate the translation equation

Substitute the given point

Solve for the translation parameter

Interpret the geometric translations

Answer: