Find and sketch through important points, i.e., x, y-intercept, vertex, etc.1.  y^2 + 6y +8x +25=0 Please explain and show detailed step by step solution.  

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neela | High School Teacher | (Level 3) Valedictorian

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To sketch   y^2 + 6y +8x +25=0

We know that this is second degree expression in y. So it is parabola.

Now put this in he standard form (y-k)^2 = 4a(x-h) , which is  astandard parabola with vertex at (h,k) and focal length a . The focus being located at (h+a , k).

y^2+6y  =  -8x-25

We add 3^2 = 9 both sides:

y^2-6y +3^2 = -8x-25+9

(y-3)^2 = - 8x-16.

(y-3)^2 =  4(-2)(x+2).

Therefore this is a parabola  with vertex at (h,k) = (-2 , 3) with focus at (-2-2 , 3) = (-4, 3) and focal length of 2.

The axis of symmetry is y = 3.

The parabola is open towards left.

The parabola intercepts x axis  at x where (0-3)^2 = -8(x+2) . Or at x= (9/-8) - 2 = -25/8. The parabola  has no y intercepts.intercepts .

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