Question 6 of 10 What can you say about the \( y \)-values of the two functions \( f(x)=-5^{x}+2 \) and \( g(x)=-5 x^{2}+2 \) ? A. \( g(x) \) has the largest possible \( y \)-value. B. \( f(x) \) and \( g(x) \) have equivalent maximum \( y \)-values. C. The maximum \( y \)-value of \( f(x) \) approaches 2 . D. \( f(x) \) has the largest possible \( y \)-value.

To analyze the \( y \)-values of the functions \( f(x) = -5^x + 2 \) and \( g(x) = -5x^2 + 2 \), let's examine each function: 1. **Function \( f(x) = -5^x + 2 \)**: - As \( x \) increases, \( 5^x \) grows exponentially, making \( -5^x \) decrease rapidly. Therefore, \( f(x) \) approaches 2 as \( x \) approaches infinity. - As \( x \) approaches negative infinity, \( 5^x \) approaches 0, so \( f(x) \) approaches 2 from below. - The maximum \( y \)-value of \( f(x) \) occurs at \( x \to -\infty \) and is equal to 2. 2. **Function \( g(x) = -5x^2 + 2 \)**: - This is a downward-opening parabola. The maximum \( y \)-value occurs at the vertex. - The vertex of a parabola given by \( ax^2 + bx + c \) is at \( x = -\frac{b}{2a} \). Here, \( a = -5 \) and \( b = 0 \), so the vertex is at \( x = 0 \). - Evaluating \( g(0) = -5(0)^2 + 2 = 2 \). Thus, the maximum \( y \)-value of \( g(x) \) is also 2. Now, let's evaluate the options: A. \( g(x) \) has the largest possible \( y \)-value. - This is false; both functions have the same maximum \( y \)-value of 2. B. \( f(x) \) and \( g(x) \) have equivalent maximum \( y \)-values. - This is true; both have a maximum \( y \)-value of 2. C. The maximum \( y \)-value of \( f(x) \) approaches 2. - This is true, but it is not the most precise statement since the maximum \( y \)-value is exactly 2. D. \( f(x) \) has the largest possible \( y \)-value. - This is false; both functions have the same maximum \( y \)-value. Based on this analysis, the best answer is: **B. \( f(x) \) and \( g(x) \) have equivalent maximum \( y \)-values.**