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) = -x² - 2
b. f(x) = 0.6x² - 2
c. f(x) = x² - 2
d. f(x) = 6x² - 2

Explanation:

To determine the equation of \( F(x) \) which is a transformation of \( G(x)=x^2 \), we analyze the graph's shape and vertical stretch/compression. The standard parabola \( y = x^2 \) has a vertex at the origin. For \( F(x) \), we check the coefficient of \( x^2 \):

• Option A: \( F(x)= -x^2 - 2 \) opens downward (negative coefficient), but the graph of \( F(x) \) (from the image) opens upward like \( G(x)=x^2 \), so A is incorrect.
• Option B: \( F(x)=0.6x^2 - 2 \). A coefficient between 0 and 1 (0.6) means the parabola is vertically compressed (wider than \( y = x^2 \)). The graph of \( F(x) \) appears wider than \( G(x)=x^2 \), and the constant term -2 shifts it down, matching the transformation.
• Option C: \( F(x)=x^2 - 2 \) has the same width as \( G(x)=x^2 \) (coefficient 1), but the graph of \( F(x) \) is wider, so C is incorrect.
• Option D: \( F(x)=6x^2 - 2 \) has a coefficient greater than 1, which would vertically stretch the parabola (narrower than \( y = x^2 \)), but the graph is wider, so D is incorrect.

Answer:

B. \( F(x) = 0.6x^2 - 2 \)