the graph shows a quadratic function.
if a system including the quadratic equation representing the parabola and a linear equation has no solution, which linear equation could be the second equation in the system?

$y = 2x + 6$
$\frac{1}{2}x = y + 4$
$2x - y = 0$
$y = 6$

Question

the graph shows a quadratic function.
if a system including the quadratic equation representing the parabola and a linear equation has no solution, which linear equation could be the second equation in the system?

$y = 2x + 6$
$\frac{1}{2}x = y + 4$
$2x - y = 0$
$y = 6$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Quadratic Functions", "Parabola Key Features", "Linear-Quadratic Systems", "Graphical Solutions" ], "new_concepts": [], "current_concepts": [ "Quadratic Functions", "Parabola Key Features", "Linear-Quadratic Systems", "Graphical Solutions" ] } ## Identify the quadratic function from the graph $$ \begin{aligned} &\text{Vertex: } (1, 6)\ &\text{y-intercept: } (0, 5)\ &y = a(x - 1)^2 + 6\ &5 = a(0 - 1)^2 + 6 \implies a = -1\ &y = -(x - 1)^2 + 6 = -x^2 + 2x + 5 \end{aligned} $$ ## Analyze the given linear equations for intersection $$ \begin{aligned} &\text{Option 1: } y = 2x + 6 \implies -x^2 + 2x + 5 = 2x + 6 \implies x^2 = -1 \quad (\text{No real solution})\ &\text{Option 2: } y = \frac{1}{2}x - 4 \implies -x^2 + 2x + 5 = \frac{1}{2}x - 4 \implies x^2 - \frac{3}{2}x - 9 = 0 \quad (\text{Real solutions exist})\ &\text{Option 3: } y = 2x \implies -x^2 + 2x + 5 = 2x \implies x^2 = 5 \quad (\text{Real solutions exist})\ &\text{Option 4: } y = 6 \implies -x^2 + 2x + 5 = 6 \implies (x-1)^2 = 0 \implies x = 1 \quad (\text{One real solution}) \end{aligned} $$ ## Determine the correct option $$ \text{The system with } y = 2x + 6 \text{ has no real solutions.} $$

(A) $y = 2x + 6$

(B) $\frac{1}{2}x = y + 4$

(C) $2x - y = 0$

(D) $y = 6$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Linear-Quadratic Systems" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the quadratic function from the graph

Analyze the given linear equations for intersection

Determine the correct option

Answer: