To determine the order of the given functions based on the smallest minimum value to the largest minimum value, we need to find the minimum values for each function [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex].
1. For the function [tex]\( f(x) = (x-3)^2 - 4 \)[/tex]:
- This is a parabola opening upwards with its vertex at [tex]\( x = 3 \)[/tex].
- The minimum value occurs at the vertex [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = (3-3)^2 - 4 = 0 - 4 = -4
\][/tex]
- Thus, the minimum value of [tex]\( f(x) \)[/tex] is [tex]\( -4 \)[/tex].
2. For the function [tex]\( g(x) \)[/tex]:
- We have the function values given in a table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
-2 & 4 \\
\hline
-1 & -2 \\
\hline
0 & -6 \\
\hline
1 & -8 \\
\hline
2 & -8 \\
\hline
3 & -6 \\
\hline
4 & -2 \\
\hline
\end{array}
\][/tex]
- The minimum value in the table is [tex]\( -8 \)[/tex].
- Thus, the minimum value of [tex]\( g(x) \)[/tex] is [tex]\( -8 \)[/tex].
3. For the function [tex]\( h(x) = -2(x-2)^2 + 1 \)[/tex]:
- This is a parabola opening downwards with its vertex at [tex]\( x = 2 \)[/tex].
- The maximum value occurs at the vertex [tex]\( x = 2 \)[/tex], but since the parabola opens downwards, this is the highest (minimum negative) point:
[tex]\[
h(2) = -2(2-2)^2 + 1 = -2 \cdot 0 + 1 = 1
\][/tex]
- Thus, the minimum value of [tex]\( h(x) \)[/tex] is [tex]\( 1 \)[/tex].
Now, we can order the functions based on their minimum values from smallest to largest:
- The minimum value of [tex]\( g(x) \)[/tex] is [tex]\( -8 \)[/tex].
- The minimum value of [tex]\( f(x) \)[/tex] is [tex]\( -4 \)[/tex].
- The minimum value of [tex]\( h(x) \)[/tex] is [tex]\( 1 \)[/tex].
Thus, the order from smallest minimum value to largest minimum value is:
[tex]\[ g(x), f(x), h(x) \][/tex]
The correct answer choice is:
[tex]\[ g(x), f(x), h(x) \][/tex]
