Question

match the given graphs as labeled to their corresponding function on the right side. (3 points) graph c graph d graph a graph b $y = 2(x - 4)^2-3$ $y = 2(x + 4)^2-3$ parent function $y=\frac{1}{2}(x - 4)^2+3$

Explanation:

Step1: Recall vertex - form of parabola

The vertex - form of a parabola is \(y=a(x - h)^{2}+k\), where \((h,k)\) is the vertex of the parabola.

Step2: Analyze the parent function

The parent function of a parabola is \(y = x^{2}\), which has a vertex at \((0,0)\). Graph A has a vertex at \((0,0)\), so Graph A corresponds to the parent function.

Step3: Analyze \(y = 2(x - 4)^{2}-3\)

For the function \(y = 2(x - 4)^{2}-3\), \(a = 2\), \(h = 4\), \(k=-3\). The vertex is \((4,-3)\). Graph D has a vertex at \((4,-3)\), so Graph D corresponds to \(y = 2(x - 4)^{2}-3\).

Step4: Analyze \(y = 2(x + 4)^{2}-3\)

For the function \(y = 2(x + 4)^{2}-3=2(x-(-4))^{2}-3\), \(a = 2\), \(h=-4\), \(k = - 3\). The vertex is \((-4,-3)\). Graph B has a vertex at \((-4,-3)\), so Graph B corresponds to \(y = 2(x + 4)^{2}-3\).

Step5: Analyze \(y=\frac{1}{2}(x - 4)^{2}+3\)

For the function \(y=\frac{1}{2}(x - 4)^{2}+3\), \(a=\frac{1}{2}\), \(h = 4\), \(k = 3\). The vertex is \((4,3)\). Graph C has a vertex at \((4,3)\), so Graph C corresponds to \(y=\frac{1}{2}(x - 4)^{2}+3\).

Answer:

Graph C: \(y=\frac{1}{2}(x - 4)^{2}+3\)
Graph D: \(y = 2(x - 4)^{2}-3\)
Graph A: Parent function
Graph B: \(y = 2(x + 4)^{2}-3\)