Question

garrett works for a company that builds parking lots. the graph shows the area of a parking lot based on the length of one side.

which equation best models the graph?

\\(a = -0.5x^2 - 69.9x + 3,263\\)
\\(a = -x^2 - 69.9x + 3,299\\)
\\(a = x^2 - 78x + 2,258\\)
\\(a = 0.5x^2 - 69.9x + 3,263\\)

Explanation:

Identify the vertex and opening direction of the parabola

The plotted points show a U-shaped curve that opens upwards.

• The minimum point (vertex) is located near \(x = 70\) with a value of \(A \approx 800\).
• Since the parabola opens upwards, the coefficient of \(x^2\) must be positive (\(a > 0\)).

Evaluate the given options based on the leading coefficient

• \(A = -0.5x^2 - 69.9x + 3,263\) (opens downwards, \(a < 0\))
• \(A = -x^2 - 69.9x + 3,299\) (opens downwards, \(a < 0\))
• \(A = x^2 - 78x + 2,258\) (opens upwards, \(a > 0\))
• \(A = 0.5x^2 - 69.9x + 3,263\) (opens upwards, \(a > 0\))

Test the vertex of the positive coefficient options

For \(A = 0.5x^2 - 69.9x + 3,263\):

$$ x_v = -\frac{b}{2a} = -\frac{-69.9}{2(0.5)} = 69.9 $$

$$ A(69.9) = 0.5(69.9)^2 - 69.9(69.9) + 3,263 = -2,443.005 + 3,263 = 819.995 \approx 820 $$

This matches the vertex in the graph at approximately \((70, 800)\).

For \(A = x^2 - 78x + 2,258\):

$$ x_v = -\frac{-78}{2(1)} = 39 $$

This vertex location (\(x = 39\)) does not match the graph's minimum near \(x = 70\).

Answer:

• (A) \(A = -0.5x^2 - 69.9x + 3,263\)
• (B) \(A = -x^2 - 69.9x + 3,299\)
• (C) \(A = x^2 - 78x + 2,258\)
• (D) \(A = 0.5x^2 - 69.9x + 3,263\) (Correct answer)