Two functions are described.

Question

f of x equals 2 x squared minus 4

g of x equals negative 2 x squared minus 4

Question
Which statement is correct?

Answer options with 4 options
A.
The graph of f of x is wider than the graph of g of x.

B.
The graph of f of x is narrower than the graph of g of x.

C.
The graph of f of x, when reflected across the line x equals negative 4, maps onto the graph of g of x.

D.
The graph of f of x, when reflected across the line y equals negative 4, maps onto the graph of g of x ​.

Answer 1

To analyze the functions \( f(x) = 2x^2 - 4 \) and \( g(x) = -2x^2 - 4 \), let's break down the characteristics of each function.

1. Differences in the Leading Coefficient:

   • The leading coefficient of \( f(x) \) is 2 (which is positive), and the leading coefficient of \( g(x) \) is -2 (which is negative).
   • A positive leading coefficient means that the parabola opens upwards (for \( f(x) \)), while a negative leading coefficient means that the parabola opens downwards (for \( g(x) \)).

2. Width of the Graph:

   • Both functions have the same absolute value of the leading coefficient (|2|), meaning they have the same width. However, since the opening direction is different, we can say:
     • The graph of \( f(x) \) is narrower than the graph of \( g(x) \) in the sense that as the value of \( x \) goes away from the vertex, the output of \( f(x) \) increases more steeply than that of \( g(x) \).

3. Reflection Across Lines:

   • To check whether reflections across particular lines will map one function onto the other, consider reflecting across the line \( y = -4 \):
     • The vertex of \( f(x) \) is at \( (0, -4) \) and that of \( g(x) \) is at \( (0, -4) \) as well, but \( g(x) \) itself is a downward-opening parabola.
     • Reflecting \( f(x) = 2x^2 - 4 \) across \( y = -4 \) effectively changes all point values because you are flipping the upward-opening graph downwards, so it will resemble \( g(x) \).

With that information, let's analyze the answer options:

• A. The graph of \( f(x) \) is wider than the graph of \( g(x) \). (Incorrect: They have the same width.)

• B. The graph of \( f(x) \) is narrower than the graph of \( g(x) \). (Incorrect: Both functions have the same width.)

• C. The graph of \( f(x) \), when reflected across the line \( x = -4 \), maps onto the graph of \( g(x) \). (Incorrect: \( x = -4 \) is a vertical line and does not map these parabolas.)

• D. The graph of \( f(x) \), when reflected across the line \( y = -4 \), maps onto the graph of \( g(x) \). (Correct: This reflection accurately describes mapping \( f(x) \) onto \( g(x) \) considering the upward and downward openings.)

Thus, the correct answer is D.