Question

find the vertex focus and directrix of parabola y=-4x^2+3x​

Answer 1

We have :-

y = -4(x² - ¾x)

=> y = -4{x² - ¾x + (⅜)²} + 4(⅜)²

=> y = -4(x - ¾)² + 9/16

=> (y - 9/16) = -4(x - ⅜)²

Let Y = (y - 9/16) and X = (x - ⅜)

=> Y = -4X²

=> X² = 4(-1/16)Y

In (X, Y) :-

Vertex : (0, 0)

Focus : (0 , -1/16)

Directrix : Y = 1/16

Now, in (x, y) :-

Vertex :

X = x - ⅜ = 0 => x = ⅜

and, Y = y - 9/16 = 0 => y = 9/16

So, vertex: (⅜, 9/16)

Focus :

X = x - ⅜ = 0 => x = ⅜

and, Y = y - 9/16 = -1/16 => x = ½

So, focus: (⅜, ½)

Directrix :

Y = y - 9/16 = 1/16 => y = ⅝

So, directrix: y = ⅝

Hope, it'll help you.....