the equation of a parabola is 12yx1248 identify the vertex f

the equation of a parabola is 12y=(x-1)^2-48. identify the vertex, focus, and directrix of the parabola. show your work

Solution

Given that

12y = ( x - 1 )² - 48

12y = x² - 2x + 1 - 48 [ since , (a - b)² = a² - 2ab + b² ]

12y =  x² - 2x - 47

y = ( x² - 2x - 47 ) / 12

y = (1/12)x² - (1/6)x - ( 47/12)

  y = (1/12)x² + (-1/6)x + ( -47/12)

This equation is in the form of standard form of parabola which is y = ax² + bx + c

a = 1/12 , b = - 1/6 , c = -47/12

Vertex :

    12y = ( x - 1 )² - 48

y = ( 1/12)(x - 1)² - 4

  y = ( 1/12)(x - 1)² + (-4)

Vertex form of parabola is , y = a(x - h)² + k

h = 1 , k = -4

Hence,

Vertex = ( h , k ) = ( 1 , -4 )

Focus : ( h , k + p ) , where p = 1/ 4a

p = 1/4(1/12)

p = 3

Hence ,

Focus = ( h , k + p )

= ( 1 , -4+3 )

= ( 1 , -1 )

Therefore,

Focus = ( 1 , -1 )

Directrix :

y = k - p

y = -4 - 3

y = -7

Therefore,

Directrix = -7