The equation of the red graph is [tex]x^{2} +2[/tex].
What is the equation of parabola whose vertex is (h, k)?
The equation of the parabola whose vertex is (h, k) is
[tex]y = a(x-h)^{2} + k[/tex]
According to the given question.
We have a graph for the function f(x) and g(x).
[tex]f(x) =x^{2}[/tex]..(i)
⇒ the equation of parabola whose vertex is (0, 0)
Since, we know the standard form of the equation of parabola is
[tex]y = a(x-h)^{2} +k[/tex]
Substitute h = 0 and k = 0 in the above equation.
⇒ [tex]y = a(x-0)^{2} +0[/tex]
⇒ [tex]y = ax^{2}[/tex]..(ii) (where, y = f(x))
On comparing the first two equations we get a = 1.
From the graph we can see that g(x) is a equation of parabola, whose vertex is (0, 2).
Therefore, the equation for red graph is given by
[tex]g(x) =a(x-0)^{2} + 2[/tex]
[tex]g(x) = 1x^{2} +2[/tex] (because a = 1)
⇒ [tex]g(x) = x^{2} +2[/tex]
Hence, option c is correct.
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