The range is all the possible y-values.
For a parabola that opens up, it is everything greater than the vertex.
If the vertex is (h,k) then the range is y >= k.
Vertex can be found using completing the square:
[tex]x^2 -10x +22 \\ \\ (x^2 -10x + (\frac{-10}{2})^2) +22 - (\frac{-10}{2})^2 \\ \\ (x^2 -10x+25) +22-25 \\ \\ (x-5)^2 - 3[/tex]
(h,k) = (5,-3) ------> Range is y >= -3
Vertex may also be found using formula [tex]h = \frac{-b}{2a} [/tex]
a = 1, b = -10, c = 22
[tex]h = \frac{10}{2} = 5[/tex]
[tex]k = 5^2 - 10(5) +22 \\ \\ k = 25 -50+22 \\ \\ k = -3[/tex]
------> Range is y >= -3
