Question

what function models the data shown on the graph?

\\(f(x) = 2(x - 5)(x - 1)\\)
\\(f(x) = -2(x - 5)(x - 1)\\)
\\(f(x) = 2(x + 5)(x + 1)\\)
\\(f(x) = -2(x + 5)(x + 1)\\)

Explanation:

Identify the data points from the graph

Using the Quadratic Modeling Interpretation knowledge point, we extract the coordinates of the plotted points from the given scatter plot:

• \(x\)-intercepts: \((1, 0)\) and \((5, 0)\)
• Vertex/maximum point: \((3, 8)\)
• Other points: \((2, 6)\) and \((4, 6)\)

Determine the factored form of the quadratic function

Using the Polynomial Roots knowledge point, since the \(x\)-intercepts are at \(x = 1\) and \(x = 5\), the quadratic function can be written in factored form:

$$ f(x) = a(x - 1)(x - 5) $$

Solve for the leading coefficient

Using the Quadratic Modeling Interpretation knowledge point, we substitute the vertex point \((3, 8)\) into the equation to find the value of \(a\):

$$
LATEXBLOCK0
$$

Thus, the function is:

$$ f(x) = -2(x - 5)(x - 1) $$

Answer:

• (A) \(f(x) = 2(x-5)(x-1)\)
• (B) \(f(x) = -2(x-5)(x-1)\) (Correct answer)
• (C) \(f(x) = 2(x+5)(x+1)\)
• (D) \(f(x) = -2(x+5)(x+1)\)