Question

the graph below can be used to solve which equation?
options:
$|x + 2| - 5 = -(x - 1)(x - 3)$
$|x - 2| - 5 = -(x - 1)(x - 3)$
$|x - 2| - 5 = -(x + 1)(x + 3)$
$|x + 2| - 5 = (x - 1)(x - 3)$

Explanation:

Step1: Analyze the absolute - value function

The general form of an absolute - value function is \(y = |x - h|-k\), where \((h,k)\) is the vertex of the V - shaped graph. From the graph, the vertex of the absolute - value (red) graph is at \((- 2,-5)\). So the equation of the absolute - value function should be \(y=|x + 2|-5\) (since \(h=-2,k = 5\) in \(y=|x - h|-k\), so \(y=|x-(-2)|-5=|x + 2|-5\)).

Step2: Analyze the quadratic function

The quadratic function (blue) is a parabola. The general form of a quadratic function in factored form is \(y=a(x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the roots. From the graph, the roots of the quadratic function are \(x = 1\) and \(x=3\), so the factored form is \(y=a(x - 1)(x - 3)\). Also, the parabola opens downwards, so \(a<0\). Let's check the value of \(a\). When \(x = 2\) (the mid - point of \(1\) and \(3\)), the vertex of the parabola. Substitute \(x = 2\) into the equation \(y=a(x - 1)(x - 3)\), we get \(y=a(2 - 1)(2 - 3)=a(1)(-1)=-a\). From the graph, at \(x = 2\), the \(y\) - value of the parabola is \(1\) (approximate). So \(-a = 1\), then \(a=-1\). So the equation of the quadratic function is \(y=-(x - 1)(x - 3)\).

Step3: Find the equation to solve

To find the equation whose solution is the intersection of the two graphs, we set the two functions equal to each other. So the equation is \(|x + 2|-5=-(x - 1)(x - 3)\).

Answer:

\(\boldsymbol{|x + 2|-5=-(x - 1)(x - 3)}\) (the first option in the list)