the graph shows the number of visitors to a museum over a seven-day period. the counts were taken twice each day.

which equation best models the graph?

$n = -2x^2 + 20x + 58$
$n = -2x^2 + 25x + 78$
$n = 2x^2 + 20x + 58$
$n = 2x^2 + 20x$

Question

the graph shows the number of visitors to a museum over a seven-day period. the counts were taken twice each day.

which equation best models the graph?

$n = -2x^2 + 20x + 58$
$n = -2x^2 + 25x + 78$
$n = 2x^2 + 20x + 58$
$n = 2x^2 + 20x$

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" ], "new_concepts": [], "current_concepts": [ "Quadratic Functions", "Parabola Key Features", "Quadratic Modeling" ] } ## Identify key features of the graph $$ \begin{aligned} &\text{y-intercept } (x = 0): N \approx 80 \ &\text{Vertex } (h, k): \text{maximum occurs near } x = 5 \text{ with } N \approx 108 \ &\text{Concavity}: \text{downward, so the coefficient of } x^2 \text{ must be negative } (a < 0) \end{aligned} $$ ## Evaluate the given options at key points $$ \begin{aligned} &\text{Option 1: } N = -2x^2 + 20x + 58 \ &\quad \text{At } x = 0: N = 58 \quad (\text{too low}) \ &\quad \text{At } x = 5: N = -2(25) + 20(5) + 58 = -50 + 100 + 58 = 108 \quad (\text{matches vertex}) \ &\text{Option 2: } N = -2x^2 + 25x + 78 \ &\quad \text{At } x = 0: N = 78 \quad (\text{matches } y\text{-intercept } \approx 80) \ &\quad \text{At } x = 5: N = -2(25) + 25(5) + 78 = -50 + 125 + 78 = 153 \quad (\text{too high}) \ &\text{Option 3: } N = 2x^2 + 20x + 58 \quad (a > 0, \text{opens upward}) \ &\text{Option 4: } N = 2x^2 + 20x \quad (a > 0, \text{opens upward}) \end{aligned} $$ ## Determine the best fitting model $$ \begin{aligned} &\text{Comparing Option 1 and Option 2 at other data points:} \ &\text{For } x = 1: \text{Graph shows } N \approx 76 \ &\quad \text{Option 1: } N = -2(1)^2 + 20(1) + 58 = 76 \quad (\text{exact match}) \ &\text{For } x = 2: \text{Graph shows } N \approx 90 \ &\quad \text{Option 1: } N = -2(4) + 20(2) + 58 = -8 + 40 + 58 = 90 \quad (\text{exact match}) \ &\text{Therefore, } N = -2x^2 + 20x + 58 \text{ is the best model.} \end{aligned} $$

(A) $N = -2x^2 + 20x + 58$

(B) $N = -2x^2 + 25x + 78$

(C) $N = 2x^2 + 20x + 58$

(D) $N = 2x^2 + 20x$

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

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QUESTION IMAGE

Question

Explanation:

Identify key features of the graph

Evaluate the given options at key points

Determine the best fitting model

Answer: