a region where theres a large amusement park is experiencing a heat wave. officials at the park created a scatter plot showing the number of visitors in the park requiring first aid during the heat wave.

which function best models the given data?

$y = -2x^2 + 20x + 60$
$y = -3x^2 + 25x + 60$
$y = -5x^2 + 50x - 13$
$y = -x^2 + 20x + 60$

Question

a region where theres a large amusement park is experiencing a heat wave. officials at the park created a scatter plot showing the number of visitors in the park requiring first aid during the heat wave.

which function best models the given data?

$y = -2x^2 + 20x + 60$
$y = -3x^2 + 25x + 60$
$y = -5x^2 + 50x - 13$
$y = -x^2 + 20x + 60$

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", "Axis of Symmetry", "Parabola Key Features" ], "new_concepts": [], "current_concepts": [ "Quadratic Functions", "Axis of Symmetry", "Parabola Key Features", "Scatter Plot Analysis" ] } ## Identify the vertex and axis of symmetry from the scatter plot $$ \begin{aligned} &\text{Vertex } (h, k) \approx (5, 116)\ &\text{Axis of symmetry: } x = -\frac{b}{2a} \approx 5 \end{aligned} $$ ## Test the axis of symmetry for each option $$ \begin{aligned} &\text{Option 1: } y = -2x^2 + 20x + 60 \implies x = -\frac{20}{2(-2)} = 5\ &\text{Option 2: } y = -3x^2 + 25x + 60 \implies x = -\frac{25}{2(-3)} \approx 4.17\ &\text{Option 3: } y = -5x^2 + 50x - 13 \implies x = -\frac{50}{2(-5)} = 5\ &\text{Option 4: } y = -x^2 + 20x + 60 \implies x = -\frac{20}{2(-1)} = 10 \end{aligned} $$ ## Evaluate the remaining candidates at the vertex $$ \begin{aligned} &\text{For } y = -2x^2 + 20x + 60 \text{ at } x = 5:\ &y = -2(25) + 20(5) + 60 = -50 + 100 + 60 = 110\ &\text{For } y = -5x^2 + 50x - 13 \text{ at } x = 5:\ &y = -5(25) + 50(5) - 13 = -125 + 250 - 13 = 112 \end{aligned} $$ ## Compare model predictions with data points $$ \begin{aligned} &\text{At } x = 1, \text{ data } y \approx 75:\ &y_1(1) = -2(1)^2 + 20(1) + 60 = 78\ &y_3(1) = -5(1)^2 + 50(1) - 13 = 32\ &\text{At } x = 9, \text{ data } y \approx 80:\ &y_1(9) = -2(81) + 20(9) + 60 = -162 + 180 + 60 = 78\ &y_3(9) = -5(81) + 50(9) - 13 = -405 + 450 - 13 = 32 \end{aligned} $$

(A) $y = -2x^2 + 20x + 60$

(B) $y = -3x^2 + 25x + 60$

(C) $y = -5x^2 + 50x - 13$

(D) $y = -x^2 + 20x + 60$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the vertex and axis of symmetry from the scatter plot

Test the axis of symmetry for each option

Evaluate the remaining candidates at the vertex

Compare model predictions with data points

Answer: