Given `y=x^2+6x+5` :
The vertex will be the minimum value of this function (the graph is a parabola opening up), and the axis of symmetry is the vertical line through the vertex.
To find the minimum take the first derivative to get `f'(x)=2x+6` .
The minimum will occur where `f'(x)=0` (extrema occur only at critical points; since polynomials are infinitely differentiable the only critical points will occur when the derivative is zero.)
2x+6=0 ==> x=-3
Thus the vertex is at (-3,f(-3)) or (-3,-4)
The axis of symmetry is x=-3
The function factors as `y=(x+1)(x+5)` so the x-intercepts are at -1 and -5. The y-intercept is at 0, so f(0)=5 implies the y-intercept is 5.
The domain is all real numbers.
The range is `y>=-4` as (-3,-4) is an absolute minimum.
To find the intervals where the function is increasing or decreasing we use the first derivative:
`2x+6>0 ==> x > -3` so the function increases on `(-3,oo)`
2x+6<0 ==> x<-3 so the function decreases on `(-oo,-3)`