Question

the graph of ( f(x) ), shown below, resembles the graph of ( g(x) = x^2 ), but it has been changed somewhat. which of the following could be the equation of ( f(x) )?
a. ( f(x) = -(x - 2)^2 )
b. ( f(x) = -(x + 2)^2 )
c. ( f(x) = x^2 - 2 )
d. ( f(x) = -x^2 - 2 )

Explanation:

Step1: Analyze the vertex of \( F(x) \)

The graph of \( G(x)=x^2 \) has its vertex at \( (0,0) \). Looking at the graph of \( F(x) \), its vertex is at \( (-2,0) \) (since it's shifted left 2 units from the origin) and it opens downward (so there's a reflection over the x - axis, which means a negative sign in front of the squared term).

Step2: Recall the vertex form of a parabola

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \( (h,k) \) is the vertex. For \( F(x) \), \( h=-2 \), \( k = 0 \), and \( a=-1 \) (because it opens downward).

Substituting these values into the vertex form: \( F(x)=-1(x - (-2))^2+0=- (x + 2)^2 \)

Answer:

B. \( F(x)=-(x + 2)^2 \)