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proof that a quadratic is a parabola

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p is focal distance, the distance from point to vertex, not the point itself. Proof that the graph of every quadratic equation/ function is a parabola: y = a x 2 + bx+ c Complete square to put into vertex form, then rearrange. Make leading coecient ! by dividing all terms on b/s by leading coecient. y a = x 2 + b a x + c a y a + ( b 2 a ) 2 = [ x 2 + b a x + ( b 2 a ) 2 ] + c a Condense perfect square. y a + ( b 2 a ) 2 = ( x + b 2 a ) 2 + c a "ubtract ( b 2 a ) 2 from both sides. y a = ( x + b 2 a ) 2 + c a − b 2 4 a 2 Multiply both sides by leading coecient. y = a ( x − ( − b 2 a ) ) 2 + [ c − b 2 4 a ] 1 a ( y − [ c − b 2 4 a ] ) = ( x − ( b 2 4 a ) ) 2 #ives connection bet$een 1 a = 4 p p = 1 4 a %x: y 2 + y − x + 1 = 0 , &nd vertex, focus, 'irectrix,

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p is focal distance, the distance from point to vertex, not the point itself.Proof that the graph of every quadratic equation/ function is a parabola:

Complete square to put into vertex form, then rearrange.Make leading coefficient 1 by dividing all terms on b/s by leading coefficient.

Condense perfect square.

Subtract from both sides.

Multiply both sides by leading coefficient.

Gives connection between

Ex: , find vertex, focus, Directrix, 1. add x to both sides. Vertex =2. ,

Vertex: 3. 4.


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