Question

PLEASE HELP!! Identify the minium value of the function

Answer 1

Because this is a positive parabola, it opens upwards, like a cup, and the vertex dictates what the minimum value of the function is.  In order to determine the vertex, I recommend completing the square.  Do that by first setting the function equal to 0 and then moving the 9 to the other side by subtraction.  So far: $x^2+6x=-9$.  Now, to complete the square, take half the linear term, square it, and add that number to both sides.  Our linear term is 6.  Half of 6 is 3 and 3 squared is 9.  So add 9 to both sides. $x^2+6x+9=-9+9$.  The right side reduces to 0, and the left side simplifies to the perfect square binomial we created while completing this process.  $(x+3)^2=0$.  Move the 0 back over and the vertex is clear now.  It is (-3, 0).  Therefore, 0 is the minimum point on your graph.  The first choice above is the one you want.