Question

the graph of f(x), shown below, resembles the graph of g(x) = x², but it has been changed somewhat. which of the following could be the equation of f(x)?
a. f(x) = - (x - 3)² - 2
b. f(x) = (x + 3)² - 2
c. f(x) = (x - 3)² - 2
d. f(x) = - (x + 3)² - 2
g(x) = x²

Explanation:

Step1: Analyze the direction of the parabola

The graph of \( G(x)=x^2 \) opens upwards. The graph of \( F(x) \) opens downwards, so there should be a negative sign in front of the squared term. This eliminates options B and C.

Step2: Analyze the horizontal shift

The vertex of \( G(x)=x^2 \) is at \( (0,0) \). The vertex of \( F(x) \) is at \( (-3, -2) \) (since it's shifted left 3 units and down 2 units). The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \( (h,k) \) is the vertex. For a horizontal shift left 3 units, \( h=-3 \), so the expression inside the square should be \( (x + 3) \) (because \( x - (-3)=x + 3 \)).

Step3: Analyze the vertical shift and direction

Since the parabola opens downwards, \( a=-1 \), and the vertical shift is down 2 units, so \( k = -2 \). Combining these, the equation should be \( F(x)=-(x + 3)^2 - 2 \), which is option D.

Answer:

D. \( F(x)=-(x + 3)^2 - 2 \)