The minimum value for y in the equation y = x² + 12x + 5 is (-6,-31). Here's how I figured it out:
Step 1: Figure out if the parabola is going up or down. If "a" is positive, the parabola will go up. If "a" is negative, the parabola will go down. In your case...
a = 1
Positive
Up
____________________________________________________________
Step 2: Find the Axis of Symmetry.
Formula: -b -12 -12
--------- = ------------ = ------------- = -6
2a 2(1) 2
The axis of symmetry is x = -6.
____________________________________________________________
Step 3: Plug in the value you found for the axis of symmetry for x into your original equation.
f(-6) = -6² + 12(-6) + 5
f(-6) = 36 - 72 + 5
f(-6) = -36 + 5
f(-6) = -31
The minimum gets written as an ordered pair: (-6,-31).
⊕ The axis of symmetry is the first number in the coordinate pair.
⊕ The answer that you get to step 3 is the second number in the pair.
____________________________________________________________
I hope that my answer helps you! This is the method that I use to solve these types of problems, but there are also other methods.