Question

How do you find the focus and directrix of y=-2x^2 +8x-15?​

Answer 1

Answer:
`\bold{focus: \bigg(2, -7 (1)/(8)\bigg), \quad directrix: y = -6 (7)/(8)}`

Explanation:

First, rearrange the equation into vertex form: y = a(x - h)² + k where

• (h, k) is the vertex
• 
  `a = (1)/(4p)`

NOTE: p is the distance from the vertex to the focus

y = -2x² + 8x - 15

y + 15 = -2x² + 8x → added 15 to both sides

y + 15 = -2(x² - 4x) → factored out -2 from the right side

y + 15 + (-2)(4) = -2(x² - 4x + 4) → completed the square

y + 7 = -2(x - 2)² → simplified

y = -2(x - 2)² - 7 → subtracted 7 from both sides

Now it is in vertex form where:

• (h, k) = (2, -7)
• a = -2 ⇒
  `-2=(1)/(4p)` ⇒
  `p=-(1)/(8)`

Focus = (2, -7 + p) → Focus = (2, -7 + (-1/8)) →
`Focus = \bigg(2, -7 (1)/(8)\bigg)`

Directrix: y = -7 - p → Directrix: y = -7 - (-1/8) →
`Directrix: y = -6 (7)/(8)`