Find the vertex and length of the latus rectum for the parabola. y=-1/2 (x-2) ^2+8

Y = - (1/2) (x-2) ² + 8

Re write it in standard form:

(y-8) = - 1/2 (x-2) ² ↔ (y-k) = a (x-h) ²

This parabola open downward (a = - 1/2 <0), with a maximum shown in vertex

The vertex is (h, k) → Vertex (2, 8)

focus (h, k + c)

a = 1/4c → - 1/2 = 1/4c → c = - 1/2, hence focus (2, 8-1/2) →focus (2,15/2)

Latus rectum: y-value = 15/2

Replace in the equation y with 15/2→→15/2 = - 1/2 (x-2) ² + 8

Or - 1/2 (x-2) ² + 8 - 15/2 = 0

Solving this quadratic equation gives x' = 3 and x" = 2, then

Latus rectum = 5