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) = 2(x + 4)^2 + 4 )
b. ( f(x) = 2(x - 4)^2 + 4 )
c. ( f(x) = -2(x + 4)^2 + 4 )
d. ( f(x) = -2(x - 4)^2 + 4 )

Explanation:

Step1: Analyze the parabola's direction

The graph of \( G(x) = x^2 \) opens upwards. The graph of \( F(x) \) opens downwards, so the coefficient of the squared term should be negative. This eliminates options A and B.

Step2: Identify the vertex form of a parabola

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

Step3: Determine the vertex coordinates

From the graph, the vertex of \( F(x) \) is at \((-4, 4)\) (since it's shifted left 4 units and up 4 units from the origin). In the vertex form, \( h = -4 \) and \( k = 4 \). Substituting into the vertex form, we get \( F(x) = a(x - (-4))^2 + 4 = a(x + 4)^2 + 4 \). Since the parabola opens downwards, \( a \) is negative. Among the remaining options (C and D), option C has \( a = -2 \) and the correct vertex shift (\( x + 4 \) for left shift of 4 units), while option D has \( x - 4 \) (which would be a right shift, incorrect).

Answer:

C. \( F(x) = -2(x + 4)^2 + 4 \)