Question

which equation best matches the graph shown below?
answer
$y = 0.5(x + 1)(x + 2)$ $y = -0.5(x + 1)(x + 2)$
$y = 0.5(x - 1)(x - 2)$ $y = -0.5(x - 1)(x - 2)$

Explanation:

Step1: Analyze the parabola's direction

The parabola opens upwards, so the coefficient of the quadratic term should be positive. This eliminates options with a negative coefficient (\(y = -0.5(x + 1)(x + 2)\) and \(y = -0.5(x - 1)(x - 2)\)).

Step2: Find the x - intercepts

The graph touches the x - axis at \(x = 1\) and \(x = 2\). For a factored form of a quadratic equation \(y=a(x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the roots (x - intercepts). Here, \(r_1 = 1\) and \(r_2=2\), so the equation should be in the form \(y=a(x - 1)(x - 2)\). Since the coefficient \(a\) is positive (from step 1) and looking at the options, \(a = 0.5\) in the option \(y = 0.5(x - 1)(x - 2)\).

Answer:

\(y = 0.5(x - 1)(x - 2)\) (the third option: \(y = 0.5(x - 1)(x - 2)\))