which graph can be used to find the real solution(s) to $3 - 4x = -3 - x^2 - 4x$?

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Graphical Solution of Systems", "Systems of Equations from Single Equations" ], "new_concepts": [], "current_concepts": [ "Graphical Solution of Systems", "Systems of Equations from Single Equations", "Quadratic Systems" ] } ## Define the system of equations Using the *Systems of Equations from Single Equations* knowledge point $$ \begin{aligned} &\text{Given equation: } 3 - 4x = -3 - x^2 - 4x\ &y_1 = 3 - 4x\ &y_2 = -3 - x^2 - 4x \end{aligned} $$ ## Analyze the linear function Using the *Graphical Solution of Systems* knowledge point $$ \begin{aligned} &y_1 = -4x + 3\ &\text{Slope } m = -4,\quad y\text{-intercept } (0, 3)\ &x\text{-intercept: } 0 = -4x + 3 \implies x = 0.75 \end{aligned} $$ ## Analyze the quadratic function Using the *Graphical Solution of Systems* knowledge point $$ \begin{aligned} &y_2 = -x^2 - 4x - 3\ &\text{Vertex } x = -\frac{b}{2a} = -\frac{-4}{2(-1)} = -2\ &y\text{-value of vertex: } y_2(-2) = -(-2)^2 - 4(-2) - 3 = 1 \implies \text{Vertex: } (-2, 1)\ &y\text{-intercept: } (0, -3)\ &x\text{-intercepts: } -x^2 - 4x - 3 = 0 \implies x^2 + 4x + 3 = 0 \implies (x+1)(x+3) = 0 \implies x = -1, -3 \end{aligned} $$ ## Identify the correct graph Using the *Graphical Solution of Systems* knowledge point $$ \begin{aligned} &\text{Line: } y = -4x + 3 \text{ (passes through } (0,3) \text{ and } (1,-1)\text{)}\ &\text{Parabola: } y = -x^2 - 4x - 3 \text{ (opens downward, vertex at } (-2,1)\text{, } x\text{-intercepts at } -1 \text{ and } -3\text{)}\ &\text{The first graph shows a line with a negative slope and a downward-opening parabola with vertex at } (-2,1). \end{aligned} $$

(A) The graph showing a line with a negative slope passing through $(0, 3)$ and a downward-opening parabola with its vertex at $(-2, 1)$.

(B) The graph showing a line with a positive slope and a downward-opening parabola in the third quadrant.

{ "elements": [ { "type": "functiongraph", "params": [ { "js": "3 - 4*x", "latex": "y = 3 - 4x" }, -10, 10 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = 3 - 4x", "withLabel": true } }, { "type": "functiongraph", "params": [ { "js": "-x*x - 4*x - 3", "latex": "y = -x^2 - 4x - 3" }, -10, 10 ], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "name": "y = -x^2 - 4x - 3", "withLabel": true } }, { "type": "point", "params": [ [-2, 1] ], "properties": { "name": "Vertex (-2, 1)", "size": 3, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [ [0, 3] ], "properties": { "name": "(0, 3)", "size": 3, "color": "#8C55F2", "withLabel": true } } ], "timestamps": [0.5, 1.0] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Graphical Solution of Systems" ] }

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Explanation:

Define the system of equations

Analyze the linear function

Analyze the quadratic function

Identify the correct graph

Answer: