Question

the function g(x) is a transformation of the parent function f(x) = x².
which of the following equations represents g(x)?
a) g(x) = (x - 3)² - 2
b) g(x) = (x - 2)² - 3
c) g(x) = (x + 2)² - 3
d) g(x) = (x - 2)²

Explanation:

Step1: Recall Vertex Form of Parabola

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

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

From the graph, the vertex of \( g(x) \) (the minimum point) is at \((2, -3)\). So \( h = 2 \) and \( k = -3 \).

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

Substituting \( h = 2 \) and \( k = -3 \) into \( y = a(x - h)^2 + k \), and since the parabola opens upward (same as parent function, so \( a = 1 \)), we get \( g(x)=(x - 2)^2 - 3 \).

Answer:

B. \( g(x)=(x - 2)^2 - 3 \)