anonymous
  • anonymous
For the function f(x) = (x − 2)2 + 4, identify the vertex, domain, and range. a. The vertex is (–2, 4), the domain is all real numbers, and the range is y ≥ 4. b. The vertex is (–2, 4), the domain is all real numbers, and the range is y ≤ 4. c. The vertex is (2, 4), the domain is all real numbers, and the range is y ≤ 4. d.The vertex is (2, 4), the domain is all real numbers, and the range is y ≥ 4.
Mathematics
katieb
  • katieb
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anonymous
  • anonymous
Michele_Laino
  • Michele_Laino
we can rewrite your parabola as below: \[\Large y - 4 = {\left( {x - 2} \right)^2}\]
Michele_Laino
  • Michele_Laino
now, if we make this change of variable: \[\Large \left\{ \begin{gathered} Y = y - 4 \hfill \\ X = x - 2 \hfill \\ \end{gathered} \right.\] where X, and Y are the new variables, then your parabola can be rewritten as below: \[\Large Y = {X^2}\]

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Michele_Laino
  • Michele_Laino
the graph of that parabola is: |dw:1439045316314:dw|
Michele_Laino
  • Michele_Laino
as we can see the vertex has the subsequent coordinates: X=0 and Y=0, so we can write: \[\Large \left\{ \begin{gathered} Y = y - 4 = 0 \hfill \\ X = x - 2 = 0 \hfill \\ \end{gathered} \right.\] so, what are x and y? Furthermore, we note that the range of our parabola is given by the subsequent equation: \[\Large Y \geqslant 0\] from which we have: \[\Large y - 4 \geqslant 0\] please solve that equation with respect to y, and you will find the requested range

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