Answer: [tex]\bold{focus: \bigg(2, -7 \dfrac{1}{8}\bigg), \quad directrix: y = -6 \dfrac{7}{8}}[/tex]
Step-by-step explanation:
First, rearrange the equation into vertex form: y = a(x - h)² + k where
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:
Focus = (2, -7 + p) → Focus = (2, -7 + (-1/8)) → [tex]Focus = \bigg(2, -7 \dfrac{1}{8}\bigg)[/tex]
Directrix: y = -7 - p → Directrix: y = -7 - (-1/8) → [tex]Directrix: y = -6 \dfrac{7}{8}[/tex]
