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# 2 more questions please

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Question 1)

Question 2)

May 5, 2018
edited by Johnnyboy  May 5, 2018

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1.   16(x - 3)  =  (y - 1)^2

The vertex of this parabola is at  ( 3, 1)

This parabola opens to the right

In the form    4p(x - 3)  = (y - k)^2......the focus is  ( 3 + p, 1)

We can solve for p as  4p  =16  ⇒   p  = 4

So....the focus  is  ( 3 + 4, 1)   =  (7,1)

The dirctrix lies 4 units to the left of the vertex and is given by    x  = ( 3 - p )  =   (3  - 4 )   =    -1

Here's a graph :  https://www.desmos.com/calculator/iwubncnnan

2.  y  = -1/6 x^2 + 7x  - 80     we need to complete the square on  x

y  = (-1/6)  [ x^2- 42x + 480 ]

Take 1/2 of 42  = 21....square it  = 441....add and subtract it within the brackets

y   =    (-1/6)  [x^2 - 42x + 441 + 480  - 441 ]     factor the frist three terms....simplify the rest

y  = (-1/6) [  (x - 21)^2  + 39 ]

y = (-1/6)(x - 21)^2  - 39/6

y  = (-1/6)(x - 21)^2  - 13/2       add 13/2 to both sides

(y + 13/2) =  (-1/6)(x - 21)^2        multiply through  by -6

-6(y + 13/2)  = (x - 21)^2

Because of the presence of a "-"   in front of the "6"......this parabola turns downward

The vertex is  ( 21, - 13/2)

The focus is  given by ( 21 , -13/2  + p )

And we can find  p as    .....  4p  = -6   ⇒  p  = -6/4  = -3/2

So....the  focus  is  (21, -13/2 - 3/2)   = ( 21 , -16/2)  = (21, -8)

The directerix is given as    y  = (-13/2 - p)   = ( -13/2 - (-3/2) )   = ( -13/2 + 3/2)  = -10/2  = -5

Here's a graph :  https://www.desmos.com/calculator/g9zfw92yos

May 5, 2018
edited by CPhill  May 5, 2018