which graph can be used to find the solution(s) to $x^2 - 4x + 4 = 2x + 1 + x^2$?

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Systems of Equations", "Graphical System Solutions" ], "new_concepts": [ "Graphical Solution of Equations" ], "current_concepts": [ "Systems of Equations", "Graphical System Solutions", "Graphical Solution of Equations" ] } ## Define the system of equations representing both sides of the equation $$ \begin{aligned} y_1 &= x^2 - 4x + 4 \ y_2 &= x^2 + 2x + 1 \end{aligned} $$ ## Analyze the properties of each quadratic function $$ \begin{aligned} y_1 &= (x - 2)^2 \implies \text{Vertex at } (2, 0), \text{ opens upward, } y\text{-intercept at } (0, 4) \ y_2 &= (x + 1)^2 \implies \text{Vertex at } (-1, 0), \text{ opens upward, } y\text{-intercept at } (0, 1) \end{aligned} $$ ## Identify the matching graph $$ \begin{aligned} &\text{The first visible graph shows a parabola with vertex at } (2, 0) \text{ opening upward, and a parabola with vertex at } (1, 0) \text{ opening downward.} \ &\text{The correct graph must show two upward-opening parabolas with vertices at } (2, 0) \text{ and } (-1, 0). \ &\text{The second, partially visible graph at the bottom shows two upward-opening parabolas matching these vertices.} \end{aligned} $$

(A) The graph showing one upward-opening parabola with vertex at $(2,0)$ and one downward-opening parabola with vertex at $(1,0)$.

(B) The graph showing two upward-opening parabolas, one with its vertex at $(2,0)$ and the other with its vertex at $(-1,0)$.

{ "elements": [ { "type": "functiongraph", "params": [ { "js": "x*x - 4*x + 4", "latex": "x^2 - 4x + 4" }, -4, 6 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = x^2 - 4x + 4", "withLabel": false } }, { "type": "functiongraph", "params": [ { "js": "x*x + 2*x + 1", "latex": "x^2 + 2x + 1" }, -4, 6 ], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "name": "y = 2x + 1 + x^2", "withLabel": false } }, { "type": "point", "params": [ [2, 0] ], "properties": { "name": "(2,0)", "size": 3, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [ [-1, 0] ], "properties": { "name": "(-1,0)", "size": 3, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [ [0.5, 2.25] ], "properties": { "name": "Intersection", "size": 4, "color": "#F2557F", "withLabel": true } } ], "timestamps": [ 0.5, 1.0, 1.5 ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Graphical System Solutions" ] }

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Question

Explanation:

Define the system of equations representing both sides of the equation

Analyze the properties of each quadratic function

Identify the matching graph

Answer: