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Help with a few more questions please :)

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Question 1) Question 2) Question 3) Thanks so much for always helping

May 5, 2018

#1
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1.  y  = -1/4x^2 + 4x  - 19

The x coordinate of the vertex  is given by   -4 / (2 * -1/4)  =  -4 / (-1/2)  =   -4 * -2    =  8

And the y coordinate is given  by :

-1/4 (8)^2  + 4(8)  - 19  =

-16  + 32   - 19

-3

So....the vertex  is  (8, -3)

Here's a graph : https://www.desmos.com/calculator/z76eahtb61

2. The center of the circle  is  ( 1, 4)   and the radius  is 8

So we have

(x - 1) ^2   + ( y - 4)^2   = 8^2        expand and simplify

x^2 - 2x + 1  + y^2  - 8y + 16  = 64

x^2 + y^2  - 2x - 8y  + 17  = 64       subtract 64 from both sides

x^2  + y^2  -2x - 8y - 47   =  0

3.  We have    y   =  1/8 x^2 + 4x + 20

We want to complete the square  on x

y = (1/8)  (x^2  + 32x + 160)

Take  1/2 of 32  = 16....square it  = 256....add and subtract it within the parentheses

y  = (1/8)  (x^2 + 32x  + 256   + 160   - 256)       factor the first three terms, simplify the last two

y = (1/8) [ (x + 16)^2  - 96 ]       apply the  1/8   over both terms in the parentheses

y  = (1/8) (x + 16)^2 - 12

(y + 12)  = (1/8)(x + 16)^2     multiply both sides by 8

8( y + 12)  = (x + 16)^2

The vertex  is at  (-16, -12)

In the form

4p ( y - k)  = (x - h)^2

4p  = 8

So....p  = 2

And the focus is given by

(-16 ,  -12 + p)   =    (-16, -12 + 2)   =   (-16, -10)

Here's the graph  :  https://www.desmos.com/calculator/cgsvcjy0ny   May 5, 2018
#2
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thank you so much CPhill you always help me so much!

Johnnyboy  May 5, 2018