Question

Part 2: Use the information provided to write the vertex form equation of each parabola.

1. Vertex: (1,4)
Focus: (7/8, 4)

2. Vertex: (5, 0)
Directrix: x = 21/4

3. Vertex: (-1, 2)
Directrix: y = 41/20

Answer 1

Answer: 1. x = -2(y - 4)² + 1

2. x = -y² + 5

3. y = -5(x + 1)² + 2

Explanation:

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

• (h, k) is the vertex
• p is the distance from the vertex to the focus

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

1)

`\text{Vertex}=(1,4)\qquad \text{Focus}:\bigg((7)/(8),4\bigg)\\\\\text{Given}: (h,k)=(1,4)\\\\\\p=focus-vertex=(7)/(8)-(8)/(8)=(-1)/(8)\\\\\\a=(1)/(4p)=(1)/(4((-1)/(8)))=(1)/(-(1)/(2))=-2`

Now input a = -2 and (h, k) = (1, 4) into the equation x = a(y - k)² + h

x = -2(y - 4)² + 1

***********************************************************************************

2)

`\text{Vertex}=(5,0)\qquad \text{Directrix}:x=(21)/(4)\\\\\text{Given}: (h,k)=(5,0)\\\\\\p=vertex-directrix=(20)/(4)-(21)/(4)=(-1)/(4)\\\\\\a=(1)/(4p)=(1)/(4((-1)/(4)))=(1)/(-1)=-1`

Now input a = -1 and (h, k) = (5, 0) into the equation x = a(y - k)² + h

x = -1(y - 0)² + 5 → x = -y² + 5

***********************************************************************************

3)

`\text{Vertex}=(-1,2)\qquad \text{Directrix}:y=(41)/(20)\\\\\text{Given}: (h,k)=(-1,2)\\\\\\p=vertex-directrix=(40)/(20)-(41)/(20)=(-1)/(20)\\\\\\a=(1)/(4p)=(1)/(4((-1)/(20)))=(1)/(-(1)/(5))=-5`

Now input a = -5 and (h, k) = (-1, 2) into the equation y = a(x - h)² + k

y = -5(x + 1)² + 2