Question

Part 1: Identify the vertex, focus, and directrix of each. Then sketch the graph.

1. y = -2(x -4)^2 - 1

2. x = (y - 1)^2 + 2

Answer 1

Answer:
`\bold{1.\quad \text{Vertex}:(4,-1)\qquad \text{Focus}:\bigg(4,-(9)/(8)\bigg)\qquad \text{Directrix}:y=-(7)/(8)}`

`\bold{2.\quad \text{Vertex}:(2,1)\qquad \text{Focus}:\bigg((9)/(4),1\bigg)\qquad \text{Directrix}:y=(7)/(4)}`

Explanation:

The vertex form of a parabola is y = a(x - h)² + k or x = a(y - k)² + h

• (h, k) is the vertex

• p is the distance from the vertex to the focus

• -p is the distance from the vertex to the directrix

`\bullet\quad a=(1)/(4p)`

1) y = -2(x - 4)² - 1 → a = -2 (h, k) = (4, -1)

`a=(1)/(4p)\quad \rightarrow \quad -2=(1)/(4p)\quad \rightarrow \quad p=-(1)/(8)\\\\\text{Focus = Vertex + p}\\\\.\qquad = (-8)/(8)+(-1)/(8)\\\\.\qquad =-(9)/(8)\qquad \rightarrow \qquad \text{Focus}=\bigg(4,-(9)/(8)\bigg)\\\\\\\text{Directrix: y=Vertex - p}\\\\.\qquad \qquad y=(-8)/(8)-(-1)/(8)\\\\.\qquad \qquad y=-(7)/(8)`

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2) x = (y - 1)² + 2 → a = 1 (h, k) = (2, 1)

`a=(1)/(4p)\quad \rightarrow \quad 1=(1)/(4p)\quad \rightarrow \quad p=-(1)/(4)\\\\\text{Focus = Vertex + p}\\\\.\qquad = (8)/(4)+(1)/(4)\\\\.\qquad =(9)/(4)\qquad \rightarrow \qquad \text{Focus}=\bigg((9)/(4),1\bigg)\\\\\\\text{Directrix: x=Vertex - p}\\\\.\qquad \qquad x=(8)/(4)-(1)/(4)\\\\.\qquad \qquad x=(7)/(4)`