Question

2.4.4 quiz: parabolas with vertices not at the origin
the vertex of this parabola is at (2, -4). which of the following could be its equation?
a. y = 2(x - 2)^2+4
b. x = 2(y + 4)^2+2
c. x = 2(y - 4)^2+2
d. y = 2(x - 2)^2-4

Explanation:

Step1: Recall vertex - form of parabola

The vertex - form of a parabola with vertex \((h,k)\) is \(y=a(x - h)^2+k\) for a parabola that opens up or down and \(x=a(y - k)^2+h\) for a parabola that opens left or right. Here the vertex is \((h,k)=(2,-4)\).

Step2: Check each option

For option A: \(y = 2(x - 2)^2+4\), the vertex is \((2,4)\) since in \(y=a(x - h)^2+k\), \(h = 2\) and \(k = 4\).
For option B: \(x=2(y + 4)^2+2\), in the form \(x=a(y - k)^2+h\), where \(h = 2\) and \(k=-4\), the vertex is \((2,-4)\).
For option C: \(x=2(y - 4)^2+2\), the vertex is \((2,4)\) as in \(x=a(y - k)^2+h\), \(h = 2\) and \(k = 4\).
For option D: \(y=2(x - 2)^2-4\), the vertex is \((2,-4)\) but from the graph, the parabola opens to the right, so it should be in the form \(x=a(y - k)^2+h\) not \(y=a(x - h)^2+k\).

Answer:

B. \(x = 2(y + 4)^2+2\)