Certainly! Let's analyze the [tex]$y$[/tex]-values of the two functions [tex]$f(x) = -5^x + 2$[/tex] and [tex]$g(x) = -5x^2 + 2$[/tex] step-by-step to determine the maximum [tex]$y$[/tex]-value for each function and compare them.
### Function [tex]\( f(x) = -5^x + 2 \)[/tex]
1. The function [tex]\( f(x) \)[/tex] involves an exponential component, [tex]\( 5^x \)[/tex].
2. The term [tex]\( 5^x \)[/tex] grows exponentially as [tex]\( x \)[/tex] increases.
3. Since there is a negative sign, [tex]\( -5^x \)[/tex] becomes a large negative value as [tex]\( x \)[/tex] increases.
4. The function [tex]\( f(x) \)[/tex] is shifted upwards by 2 units due to the [tex]\( +2 \)[/tex] term.
5. To find the maximum [tex]\( y \)[/tex]-value, look at behavior when [tex]\( x \)[/tex] is small. When [tex]\( x \)[/tex] is zero:
[tex]\[
f(0) = - 5^0 + 2 = -1 + 2 = 1
\][/tex]
Since [tex]\( -5^x \)[/tex] decreases as [tex]\( x \)[/tex] increases, the highest value of [tex]\( f(x) \)[/tex] is when [tex]\( x = 0 \)[/tex], giving [tex]\( f(0) = 1 \)[/tex].
### Function [tex]\( g(x) = -5x^2 + 2 \)[/tex]
1. The function [tex]\( g(x) \)[/tex] is a quadratic function with a downward parabolic shape because of the negative coefficient [tex]\( -5 \)[/tex].
2. The term [tex]\( -5x^2 \)[/tex] indicates the parabola opens downward.
3. The maximum value of a parabola [tex]\( ax^2+bx+c \)[/tex] occurs at the vertex. Since this is a simple quadratic term without linear coefficient (no [tex]\( bx \)[/tex]), the vertex is at [tex]\( x = 0 \)[/tex].
4. Evaluating [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = -5(0)^2 + 2 = 2
\][/tex]
Hence, the maximum [tex]\( y \)[/tex]-value of [tex]\( g(x) \)[/tex] is 2, which occurs at [tex]\( x = 0 \)[/tex].
### Comparing the Maximum [tex]\( y \)[/tex]-Values:
- The maximum [tex]\( y \)[/tex]-value for [tex]\( f(x) = -5^x + 2 \)[/tex] is 1.
- The maximum [tex]\( y \)[/tex]-value for [tex]\( g(x) = -5x^2 + 2 \)[/tex] is 2.
Therefore, the function [tex]\( g(x) \)[/tex] has the largest possible [tex]\( y \)[/tex]-value among the two functions.
Thus, the correct answer is:
[tex]\[
\boxed{C. \; g(x) \; has \; the \; largest \; possible \; y \text{-value}.}
\][/tex]
