​Quadratic Relationships

Standard form has many parts to it, you have Zeroes, the axis of symmetry, optimal value and the completing the square to turn into vertex form.

 

So for Zeroes, in standard form it is possible for you to make it into factored form but, this will not wok all the time so you can just use the Quadratic Formula. So in this case, y = ax2 + bx + c will become b+-√b²-4(a)(c).

                     2(a)

 

Next you have the axis of symmetry, so in standard form to find the axis of symmetry you have to use the formula (-b/2a). To use this what you would do is take the b value out of ax2 + bx + c and make it negative then you would put in the value for "a" in 2a. For example if you had 

y = 3x2 + 6x + 12, to find the axis of symmetry you would do -6 / 2 (3) and then you would have your A.O.S.

 

Now you have optimal Value, to find the optimal value you would need to find your A.O.S first. After youve done that then you can take the axis of symmetry and substitue it for x in the original equation and solve for y.

 

Lastly you have completing the square to turn it into vertex form. Now to do this you would have to make your standard form equation have a square "y=a(x-h)²+k" because vertex form looks like a square. 

 

For Example, y = 2x2 + 60x - 3, find the vertex. 

y= 2 ( x + 30x) -3

y= 2 ( x + 30x + 225 - 225) -3

y= 2 ( (x + 15)2 )- 255) -3

y= 2 ( x + 15)2 - 453