Question

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What is the correct standard form of the equation of the parabola?

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Answer 1

`(x-2)^2=4(y+5)`

Explanation:

The standard form of a parabola with a vertical axis of symmetry is given by the formula:

`\boxed{(x-h)^2=4p(y-k)}`

where:

• p ≠ 0
• Vertex = (h, k)
• Focus = (h, k+p)
• Directrix: y = (k - p)
• Axis of symmetry: x = h

From inspection of the given graph:

• Vertex: (2, -5)
• Focus: (2, -4)
• Directrix: y = -6

Therefore:

• h = 2
• k = -5

Use the formula for the directrix to find the value of p:

`\implies k - p = -6`

`\implies -5 - p = -6`

`\implies -5 - p +5= -6+5`

`\implies -p = -1`

`\implies p=1`

Substitute the found values of h, k and p into the formula:

`\implies (x-2)^2=4(1)(y-(-5))`

`\implies (x-2)^2=4(y+5)`

Therefore, the correct standard form of the equation of the parabola is:

`\boxed{(x-2)^2=4(y+5)}`