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1735
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What is the focus of the parabola?

y=−14x2−2x−2

Enter your answer in the boxes.

(__,__)

 

 

The equation of a parabola is y=18x2−4x+41 .

What is the equation of the directrix?

 

y = 14

y = 11

​y = 7​

​y = 2​

 

The equation of a parabola is 132(y−2)2=x−1 .
What are the coordinates of the focus?

(9, 2)

(−7, 2)

(1, 10)

(1, −6)

 Jun 6, 2017
 #1
avatar+98123 
+1

 

y  = -14x^2 - 2x - 2

 

We want to get to the form     4p (y - k)  = ( x - h)^2  

 

Where (h, k)  is the vertex  and p is the distance from the vertex to the focus

 

We can factor this as

 

y =  -14 [x^2  + (1/7)x + (1/7) ]

 

Complete the square inside the brackets

Take (1/2)of (1/7)  = (1/14)....square this  = 1/196  ...add it and subtract it

So we have

 

y  =  -14 [ x^2 +(1/7)x + 1/196 +  (1/7) - (1/196) ]

Factor the first three terms  and simplify the last two.....so we have

 

y  =  -14 [  (x + 1/14)^2  + 27/196]      simplify

 

y = -14 ( x + 1/14)^2 - 27/14      add 27/14 to both sides

 

(y +27/14)  =  -14 ( x + 1/14)^2   multiply both sides by -1/14

 

(-1/14)(y + 27/14)  = ( x + 1/14)^2

 

Since  -1/14  =  4p....divide both sides by 4.....then  p  =  - 1/56

 

So...   the  vertex  is  (h, k) = ( -1/14, -27/14 )

 

And the focus  is given by ( h , k + p)  =  

 

( -1/14, - 27/14 - 1/56)  =

 

(- 1/14,  - 109/56 )

 

 

Here's the graph :  https://www.desmos.com/calculator/moljoeobcv

 

 

 

cool cool cool

 Jun 6, 2017
edited by CPhill  Jun 6, 2017
 #2
avatar+98123 
+1

 

The equation of a parabola is y=18x^2−4x+41 .

What is the equation of the directrix?

 

We want the form  4p(y - k) = (x - h)^2      where (h, k) is the  vertex  and p is the distance between the vertex and the directrix

 

Factor out 18

 

y  =  18 ( x^2 - (2/9)x  +  41/18 )

 

Complete the square  inside the parentheses

Take (1/2) of (2/9) = (1/9) .....square it  = (1/81)....add and subtract it

 

y =  18 [  x^2 - (2/9)x + 1/81 +  41/18 - 1/81 ]   

factor the first three terms, simplify the last two

 

y  = 18 [ (x - 1/9)^2  + 367/162]    simplify

 

y  =  18 ( x - 1/9)^2 +  367/9     

subtract  367/9 from both sides

 

(y - 367/9) =  18 ( x - 1/9)^2      divide both sides by 18

 

(1/18) (y - 367/9)  = ( x - 1/9)^2

 

Since  4p  = 1/18,   divide both sides by 4....so p  =  1/72

 

The vertex  is  ( 1/9, 367/9)

 

And  the  equation of the directrix  is   y  = 367/9 - p    →  y  = 367/9 - 1/72  →

 

y = 2935 / 72

 

 

Here's the graph : https://www.desmos.com/calculator/qjwolpipur

 

 

cool cool cool

 Jun 6, 2017
edited by CPhill  Jun 6, 2017
 #3
avatar+98123 
+1

The equation of a parabola is 132(y−2)^2 = x−1  .
What are the coordinates of the focus?

 

We want the  form

 

4p ( x - h)  = (y - k)^2     wher (h, k) is the vertex  and p  is the distance between the vertex and the focus

 

132 (y - 2)^2  =  (x - 1)     rearrange as

 

(x - 1)  =  132 ( y - 2)^2      divide both sides by 132

 

(1/132) ( x - 1)  = ( y - 2)^2

 

4p  =  (1/132)     divide both sides by 4   and p  =  1/528

 

And the focus  is given by

 

( h + p , k)   =  ( 1 + 1/528 , 2)   =  ( 529/528, 2 )

 

Here's the graph : https://www.desmos.com/calculator/cqctvmc09d

 

 

cool cool cool

 Jun 6, 2017

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