Question 422025
  <pre><font face = "Tohoma" size = 4 color = "indigo"><b>
Hi
Using the vertex form of a parabola, {{{y=a(x-h)^2 +k}}} where(h,k) is the vertex
for what values of k will the function f(x)=x^2 +6x+k
f(x)=x^2 +6x+k = (x+3)^2 -9 +k     |completing the square
Note parabola opens upward(coefficient of the squared term >0)
a) cut the a-axis twice (-9+k)<0     or k < 9  Vertex below the x-axis
b) touch the x-axis (-9+k) = 0       or k = 9
c) have no x- intercepts (-9+k)> 0   or k > 9  Vertex above the x-axis
 
{{{drawing(300,300,   -6, 6, -6, 6,  blue(line(-3,6,-3,-6))  , grid(1),
graph( 300, 300, -6, 6, -6, 6, 0 ,(x+3)^2 - 2,(x+3)^2,(x+3)^2+2 ))}}}