Given the quadratic equation y=x^2+6x+5 find the vertex, axis of symmetry, intercepts, domain,range,interval where the function is increasing and decreasing  

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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)`

The graph:

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