Question

put the following functions in order of smallest minimum value to largest minimum value.
$f(x) = (x - 3)^2 - 8$

$x$	$g(x)$
$-3$	$1$
$-2$	$-2$
$-1$	$-3$
$0$	$-2$
$1$	$1$
$2$	$6$

graph of $h(x)$

Explanation:

To solve the problem of ordering the functions by their minimum values, we analyze each function:

Step 1: Analyze \( f(x) = (x - 3)^2 - 8 \)

This is a quadratic function in vertex form \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. For \( f(x) = (x - 3)^2 - 8 \), the vertex is \( (3, -8) \). Since the coefficient of \( (x - 3)^2 \) is positive, the parabola opens upward, so the minimum value of \( f(x) \) is \( -8 \).

Step 2: Analyze \( g(x) \) from the table

We look at the values of \( g(x) \) in the table:

\( x \)	\( g(x) \)
-3	1
-2	-2
-1	-3
0	-2
1	1
2	6

The values of \( g(x) \) first decrease, reach a minimum, then increase. The minimum value of \( g(x) \) is \( -3 \) (at \( x = -1 \)).

Step 3: Analyze \( h(x) \) from the graph

The graph of \( h(x) \) is a parabola opening upward (since the arms of the parabola go up). The vertex (minimum point) of \( h(x) \) appears to be at \( (0, 0) \) (from the graph, it seems to touch the origin). So the minimum value of \( h(x) \) is \( 0 \).

Now we compare the minimum values:

• Minimum of \( f(x) \): \( -8 \)
• Minimum of \( g(x) \): \( -3 \)
• Minimum of \( h(x) \): \( 0 \)

Ordering from smallest to largest minimum value: \( f(x) \), \( g(x) \), \( h(x) \)

Answer:

The order from smallest minimum value to largest minimum value is \( f(x) \), \( g(x) \), \( h(x) \)