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 of the parabola. For the parent function \( f(x)=x^2 \), the vertex is at \((0, 0)\).

Step2: Determine the vertex of \( g(x) \)

From the graph, the vertex of \( g(x) \) is at \((- 4, 6)\) (since it is shifted 4 units to the left and 6 units up from the origin). In the vertex form \( g(x)=(x - h)^2 + k \), if the vertex is \((h,k)\), here \( h=-4 \) and \( k = 6 \).

Step3: Substitute \( h \) and \( k \) into vertex form

Substitute \( h=-4 \) and \( k = 6 \) into \( g(x)=(x - h)^2 + k \). We get \( g(x)=(x-(-4))^2+6=(x + 4)^2+6 \).

Answer:

\( g(x)=(x + 4)^2+6 \) (the option corresponding to this equation)