Question

which equation best matches the graph shown below?

answer
\\( y = -2(x - 3)(x - 4) \\) \\( y = 2(x + 3)(x + 4) \\)
\\( y = -2(x + 3)(x + 4) \\) \\( y = 2(x - 3)(x - 4) \\)

Explanation:

Step1: Identify x - intercepts

The graph intersects the x - axis at \(x = 3\) and \(x=4\). For a quadratic equation in factored form \(y=a(x - r_1)(x - r_2)\), the roots (x - intercepts) are \(r_1\) and \(r_2\). So the equation should have factors \((x - 3)\) and \((x - 4)\), eliminating options with \((x + 3)\) and \((x + 4)\) (i.e., \(y = 2(x + 3)(x + 4)\) and \(y=-2(x + 3)(x + 4)\)).

Step2: Determine the sign of 'a'

The parabola opens downward. For a quadratic function \(y = a(x - r_1)(x - r_2)\), if \(a<0\), the parabola opens downward; if \(a > 0\), it opens upward. So \(a\) should be negative. Among the remaining options (\(y=-2(x - 3)(x - 4)\) and \(y = 2(x - 3)(x - 4)\)), the one with \(a=-2\) (i.e., \(y=-2(x - 3)(x - 4)\)) has a negative leading coefficient, so the parabola opens downward.

Answer:

\(y=-2(x - 3)(x - 4)\)