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1. The general form of an parabola is 3x^2 + 24x − 2y + 52 = 0 .

What is the standard form of the parabola?

Enter your answer by filling in the boxes. Enter any fractions in simplest form.

2.The directrix of a parabola is x = 4. Its focus is (2,6) .

What is the standard form of the parabola?

Nov 14, 2019

#1
+1

Here is ONE standard form

3x^2+24x-2y+52    = 0       divide by 3

x^2 + 8x -2/3 y + 52/3         complet 'x' square

(x+4)^2    = 16    +2/3 y -52/3

(x+4)^2 = 2/3 y -4/3

(x+4)^2 =  2/3 (y - 2)

Nov 14, 2019
#2
+2

3x^2 + 24x  - 2y + 52  =  0       rearrange as

2y  =  3x^2 + 24x  + 52           divide both sides by  2

y = (3/2)x^2 + 12x + 26          factor out the (3/2)

y  =  (3/2)  [ x^2  + 8x + 52/3 ]

Complete the square on x........take 1/2  of 8  = 4  ......square it   =  16.....add and subtract within the parentheses

y = (3/2) [ x^2 + 8x + 16 + 52/3  - 16 ]

Factor the first three terms in the parentheses  and simplify the last two terms

y =  (3/2) [ (x + 4)^2  + 4/3 ]

Distribute the (3/2)  over the terms in the parentheses

y  = (3/2) (x + 4)^2  + 2    or

(y - 2)  = (3/2) (x + 4)^2   Nov 14, 2019
#3
+2

Second one

The vertex will be at   ([4 + 2] / 2, 6 )  =   ( 3, 6) =  (h, k)

This parabola opens to the left   because the directrix is to the right of the focus

The form is

-4a ( x - h)  = (y - k)^2

a = the distance between the vertex and focus  =  1

So  we have that

-4 (1) (x - 3)  = (y - 6)^2

-4 (x - 3)  = (y - 6)^2

Here's the graph  :  https://www.desmos.com/calculator/9wy9pmdttx   Nov 14, 2019