write an equation for the function show below, which is a transformed version of $f(x) = x^2$ that has been…

write an equation for the function show below, which is a transformed version of $f(x) = x^2$ that has been shifted vertically and horizontally.\n$g(x) = \\square$

write an equation for the function show below, which is a transformed version of $f(x) = x^2$ that has been shifted vertically and horizontally.\n$g(x) = \\square$

Answer

Explanation:

Step1: Identify Vertex of Parabola

The parent function is ( f(x) = x^2 ) with vertex at ((0,0)). The graph shown has its vertex at ((-3, 2)).

Step2: Recall Vertex Form of Quadratic

The vertex form of a quadratic function is ( g(x) = a(x - h)^2 + k ), where ((h, k)) is the vertex. For ( f(x) = x^2 ), ( a = 1 ), and since there's no vertical stretch/compression (shape same as ( x^2 )), ( a = 1 ).

Step3: Substitute Vertex into Form

Here, ( h = -3 ), ( k = 2 ). Substituting into ( g(x) = (x - h)^2 + k ), we get ( g(x) = (x - (-3))^2 + 2 = (x + 3)^2 + 2 ).

Answer:

( (x + 3)^2 + 2 )