Question

what is the equation of the translated function, g(x), if f(x) = x²? g(x) = (x - 4)² + 6 g(x) = (x + 6)² - 4 g(x) = (x - 6)² - 4 g(x) = (x + 4)² + 6

Explanation:

Step1: Recall Vertex Form of Parabola

The vertex form of a parabola is \( g(x) = (x - h)^2 + k \), where \((h, k)\) is the vertex. For \( f(x)=x^2 \), the vertex is \((0, 0)\).

Step2: Determine Vertex of \( g(x) \)

From the graph, the vertex of \( g(x) \) is at \((-4, 6)\) (since it's shifted left 4 units and up 6 units from \( f(x) \)'s vertex \((0,0)\)).

Step3: Substitute \( h \) and \( k \) into Vertex Form

In \( g(x) = (x - h)^2 + k \), substitute \( h = -4 \) and \( k = 6 \). So \( g(x) = (x - (-4))^2 + 6=(x + 4)^2 + 6 \).

Answer:

\( g(x) = (x + 4)^2 + 6 \) (the fourth option: \( g(x) = (x + 4)^2 + 6 \))