+0  
 
0
192
3
avatar+98 

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)

CrazyDaizy  Jun 6, 2017
Sort: 

3+0 Answers

 #1
avatar+79894 
+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

CPhill  Jun 6, 2017
edited by CPhill  Jun 6, 2017
 #2
avatar+79894 
+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

CPhill  Jun 6, 2017
edited by CPhill  Jun 6, 2017
 #3
avatar+79894 
+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

CPhill  Jun 6, 2017

6 Online Users

avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details