Omg ! the function f(x) = 4(x-7)^2-22 has been rewritten using the completing-the-square method. the function g(x) = 6x^2+12x+14 . which vertex for each function has a minimum or a maximum? what is the vertex for each function? explain your reasoning for each function

Answers:

Both functions have minima at the vertex.  The vertices are (7,-22) and (-1, 8) respectively.

Step-by-step explanation:

Hello!  Let's look at the general form for the vertex equation of a parabola:

y-k = a(x-h)^2, or (alternativelyl) y = a(x-h)^2 + k, where (h,k) is the vertex.

Comparing the given f(x) to y = a(x-h)^2 + k, we see immediately that h = 7 and k = -22, so that the vertex is (7,-22).  Since that coefficient 4 is positive, we know that the graph of this parabola opens up and that the vertex represents the minimum of f(x).

We could complete the square to rewrite g(x) in vertex form, or we could recall that the equation of the axis of symmetry is x = -b/ (2a).  Here, a = 6 and b = 12, so the axis of symmetry is x = -12 / (2*6), or x = -1.  Using synthetic division to evaluate g(x) at this x = -1, we get y = g(-1) = 8.  Thus, the vertex of g(x) is (-1, 8).  This represents the minimum value of g(x).

Both functions have minima at the vertex.

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step-by-step explanation:

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