which graph shows solving $2x^2 - 8x + 5 = -x^2 + 4$ by graphing a system?

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Systems of Equations", "Graphical Intersection of Systems" ], "new_concepts": [], "current_concepts": [ "Systems of Equations", "Graphical Intersection of Systems", "Quadratic Functions" ] } ## Identify the system of equations $$ \begin{aligned} y_1 &= 2x^2 - 8x + 5 \ y_2 &= -x^2 + 4 \end{aligned} $$ ## Analyze the properties of each function $$ \begin{aligned} &\text{For } y_1 = 2x^2 - 8x + 5:\ &\text{Vertex } x = -\frac{-8}{2(2)} = 2\ &y_1(2) = 2(4) - 16 + 5 = -3 \implies \text{Vertex at } (2, -3)\ &\text{Opens upward } (a = 2 > 0)\ \ &\text{For } y_2 = -x^2 + 4:\ &\text{Vertex at } (0, 4)\ &\text{Opens downward } (a = -1 < 0) \end{aligned} $$ ## Match with the correct graph $$ \begin{aligned} &\text{Graph 1 (Left): Shows one upward-opening parabola with vertex at } (2, -3) \text{ and one downward-opening parabola with vertex at } (0, 4).\ &\text{Graph 2 (Middle): Shows two upward-opening parabolas.}\ &\text{Graph 3 (Right): Shows two downward-opening parabolas.}\ &\text{Therefore, the first graph (left) is the correct representation.} \end{aligned} $$

(A) The first graph (left), showing an upward-opening parabola with vertex at $(2, -3)$ and a downward-opening parabola with vertex at $(0, 4)$.

(B) The second graph (middle), showing two upward-opening parabolas.

(C) The third graph (right), showing two downward-opening parabolas.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Graphical Intersection of Systems" ] }

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Question

Explanation:

Identify the system of equations

Analyze the properties of each function

Match with the correct graph

Answer: