Question

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

Answer 1

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

Step-by-step explanation:

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

• (h, k) is the vertex
• $a = \dfrac{1}{4p}$

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              ⇒    $-2=\dfrac{1}{4p}$     ⇒    $p=-\dfrac{1}{8}$

Focus = (2, -7 + p)  →  Focus = (2, -7 + (-1/8))   →  $Focus = \bigg(2, -7 \dfrac{1}{8}\bigg)$

Directrix: y = -7 - p   →  Directrix: y = -7 - (-1/8)   →  $Directrix: y = -6 \dfrac{7}{8}$