Convert \( y = x^2 + 64x + 12 \) into graphing form.

T o co n v er t t h e s t an d a r d f or m y = a x 2 + b x + c o f a f u n c t i o n in t o v er t e x f or m y = a ( x − h ) 2 + k , w e ha v e t o w r i t e t h e e q u a t i o n in t h e co m pl e t e s q u a re f or m an d v er t e x ( h , k ) i s g i v e n b y :
h = 2 a − b ​ , k = c − 4 a b 2 ​ y = a ( x − h ) 2 + k
opens up for a > 0, and down for a < 0
y = x 2 + 64 x + 12 a = 1 , b = 64 c = 12 h = 2 − 64 ​ = − 32 k = 12 − 4 6 4 2 ​ = 12 − 4 4096 ​ = 12 − 1024 = − 1012 y = ( x + 32 ) 2 − 1012 T hi s m e an s t h e v er t e x o f t h e p a r ab o l a i s a t t h e p o in t ( − 32 , − 1012 ) an d t h e p a r ab o l a i s co n c a v e u p .

Answered by Lilith | 2024-06-10

To convert y = x 2 + 64 x + 12 into vertex form, we complete the square and find that it can be expressed as y = ( x + 32 ) 2 − 1012 . This indicates the vertex is at the point ( − 32 , − 1012 ) . The parabola opens upwards since the coefficient of x 2 is positive.
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Answered by Lilith | 2024-12-23