good to know you've got your basics straightened out
let's work with a basic equation for example:
y = 1x^2 +4x +9
Step 1: combine your ax^2 +bx terms together by factoring; HINT: get 1x^2 inside the brackets:
y = 1(x^2 +4x) + 9
---can be just a factor of 1 to help isolate!
Step 2: look at your 'new' bx term. take b. half it. square it. ADD and SUBTRACT.
y = 1(x^2 +4x +2^2 - 2^2) + 9
y = 1(x^2 +4x +4 - 4) + 9
----simplify
Step 3: Kick out your SUBTRACTed term from the brackets by multiplying it by the brackets' coefficient.
y = 1(x^2 +4x +4) + 1(-4) + 9
y = 1(x^2 +4x +4) - 4 + 9
----------simplify
y = 1(x+2)(x+2) + 5
---simplify, and factor the now-perfect trinomial
y = 1(x+2)^2 + 5
y = (x+2)^2 +5
SO our vertex here is (-2, 5).... axis of symmetry: x = -2
Here's why I put the '1' part in for the factoring at the beginning: let's take the following equation instead:
y = 2x^2 +4x + 9
y = 2(x^2 +2x) + 9
---factor to get 1x^2 inside the brackets
y = 2(x^2 +2x +1^2 - 1^2) + 9
-half of new b = 1, squared is 1
y = 2(x^2 +2x +1 - 1) + 9
y = 2(x^2 +2x +1) + 2(-1) + 9
-kick out SUBTRACTed term by multiplication
y = 2(x^2 +2x +1) - 2 + 9
y = 2(x+1)(x+1) + 7
--simplify and factor now-perfect trinomial
y = 2(x+1)^2 + 7
so our vertex here is (-1, 7) and our axis of symmetry... x = -1
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