Question

which equation best matches the graph shown below?
answer
\\( y = -0.7(x + 6)(x + 7) \\) \\( y = 0.7(x + 6)(x + 7) \\)
\\( y = 0.7(x - 6)(x - 7) \\) \\( y = -0.7(x - 6)(x - 7) \\)

Explanation:

Step1: Identify x-intercepts

The graph intersects the x - axis at \(x=-7\) and \(x = - 6\). For a quadratic equation in factored form \(y=a(x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the roots, the roots here are \(r_1=-7\) and \(r_2=-6\). So the equation should be in the form \(y=a(x + 7)(x + 6)\) (since \(x-(-7)=x + 7\) and \(x-(-6)=x + 6\)).

Step2: Determine the sign of 'a'

The parabola opens downward, which means the coefficient \(a\) of the quadratic term (when expanded) is negative. So \(a<0\). Among the given options, the equation \(y=-0.7(x + 6)(x + 7)\) has \(a=-0.7\) (negative) and the roots at \(x=-6\) and \(x=-7\) (since when \(y = 0\), \(x+6 = 0\) or \(x + 7=0\)).

Answer:

\(y=-0.7(x + 6)(x + 7)\)