6) $f(x)=(x + 5)^2+3$\n$a =$\n$h =$\n$k =$\nopens (up or down):\nmaximum or minimum:\nvertex:\naxis of…

Explanation:

Step1: Identify the form of the quadratic function

The function ( f(x)=(x + 5)^{2}+3 ) is in vertex form ( f(x)=a(x - h)^{2}+k ). Here, ( a = 1 ), ( h=-5 ), ( k = 3 ).

Step2: Determine the direction it opens

Since ( a=1>0 ), the parabola opens up.

Step3: Find the maximum or minimum

Because it opens up, it has a minimum. The minimum value is ( k = 3 ).

Step4: Identify the vertex

The vertex of the parabola in vertex form ( f(x)=a(x - h)^{2}+k ) is ( (h,k) ). So the vertex is ( (-5,3) ).

Step5: Find the axis of symmetry

The axis of symmetry for a parabola in the form ( f(x)=a(x - h)^{2}+k ) is ( x = h ). So the axis of symmetry is ( x=-5 ).

Step6: Determine the domain

For any quadratic function, the domain is all real numbers. So ( x\in(-\infty,\infty) ).

Step7: Determine the range

Since the parabola opens up and has a minimum value of ( y = 3 ), the range is ( y\in[3,\infty) ).

Answer:

Opens (up or down): up
Maximum or Minimum: Minimum at ( y = 3 )
Vertex: ( (-5,3) )
Axis of Symmetry: ( x=-5 )
Domain: ( (-\infty,\infty) )
Range: ( [3,\infty) )