I need help with #11 and #12

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  11. y = -1/4x² +1

  12. y = 1/6(x -3)² +5/2

Step-by-step explanation:

Since 9-12 are all the same, we can solve the problem generically and plug in the given numbers. Let the focus be (a, b) and the directrix be y=d.

The vertex is halfway between the focus and directrix, so its coordinates will be (a, (b+d)/2). The distance from focus to directrix is b-d. Given these conditions, the equation of the parabola is ...

  y = 1/(2(b-d))(x -a)² +(b+d)/2

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11. For (a, b) = (0, 0) and d = 2, the equation is ...

  y = 1/(2(0-2))(x -0)² +(0+2)/2

  y = -1/4x² +1

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12. For (a, b) = (3, 4) and d = 1, the equation is ...

  y = 1/(2(4-1))(x -3)² +(4+1)/2

  y = 1/6(x -3)² +5/2

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The attachments show the graphs of these. You can verify the graph by noticing ...

  a) the vertex is halfway between the focus and directrix

  b) a horizontal line from the focus to the curve is the same length as a vertical line from the directrix to the same point on the curve

g(x) = x² + 2

step-by-step explanation:

it is given that the graphs are the same shape.

the only difference is that the red graph is 2 units higher than the blue graph.

that is to say that the red graph is simply the blue graph that has been moved vertically by 2 units in the positive y direction.

hence

g(x) = f(x) + 2

or

g(x) = x² + 2

details

step-by-step explanation: