Question

four quadratic functions are represented below:
( a(x) = (x - 3)^2 - 7 )
( b(x) = x^2 + 6x + 9 )
graph of ( h(x) ) (a parabola)
table for ( k(x) ):

( x )	( k(x) )
3	4
2	6
1	4
0	1

which function has the smallest minimum value?
(options include iii etc.)

Explanation:

Step1: Analyze Function I

The function is \( a(x)=(x - 3)^2 - 7 \). For a quadratic function in vertex form \( y = a(x - h)^2 + k \), the vertex is \((h,k)\). Here, \( h = 3 \), \( k=-7 \), so the minimum value (since \( a = 1>0 \), it opens upward) is \( -7 \).

Step2: Analyze Function II (Graph)

The graph of \( b(x) \) is a parabola opening upward. From the graph, the vertex (minimum point) has a \( y \)-coordinate (minimum value) of \( -4 \) (estimating from the grid, the lowest point is at \( y=-4 \)).

Step3: Analyze Function III

The function is \( c(x)=x^2 + 6x + 9 \). We can rewrite it in vertex form by completing the square: \( x^2 + 6x + 9=(x + 3)^2 \). So the vertex is \((-3,0)\), and the minimum value (since \( a = 1>0 \)) is \( 0 \).

Step4: Analyze Function IV (Table)

For the table of \( d(x) \), we can find the vertex. The table values are symmetric around \( x = 2 \) (since \( d(4)=1 \), \( d(0)=1 \); \( d(3)=4 \), \( d(1)=4 \); \( d(2)=6 \)). So the vertex is at \( (2,6) \), and the minimum value is \( 6 \) (since the parabola opens upward, as the \( y \)-values increase away from \( x = 2 \)).

Step5: Compare Minimum Values

Now we compare the minimum values: Function I: \( -7 \), Function II: \( -4 \), Function III: \( 0 \), Function IV: \( 6 \). The smallest minimum value is \( -7 \), which belongs to Function I. Wait, but the original problem's option shown is "I" maybe? Wait, let's re - check. Wait, maybe I misread the function I. Wait, \( a(x)=(x - 3)^2-7 \), so minimum at \( y=-7 \). Function II's graph: looking at the grid, the lowest point is at \( y = - 4 \) (the vertex is at \( (0,-4) \)? Wait, maybe my initial analysis of Function II was wrong. Wait, the graph of \( b(x) \): the parabola has its vertex at \( (0,-4) \)? Wait, no, looking at the graph, the \( y \)-axis is vertical. The graph crosses the \( y \)-axis at \( (0,-3) \)? Wait, maybe I made a mistake. Wait, let's re - evaluate.

Wait, Function I: \( a(x)=(x - 3)^2-7 \), minimum value \( -7 \).

Function II: The graph of \( b(x) \): the vertex (minimum point) is at \( (0,-4) \) (since the parabola is symmetric about the \( y \)-axis, and the lowest point is at \( y=-4 \)).

Function III: \( c(x)=x^2 + 6x + 9=(x + 3)^2 \), minimum at \( y = 0 \).

Function IV: The table for \( d(x) \): when \( x = 2 \), \( d(x)=6 \), and as \( x \) moves away from 2, \( d(x) \) increases, so minimum at \( y = 6 \).

So comparing \( -7 \) (Function I), \( -4 \) (Function II), \( 0 \) (Function III), \( 6 \) (Function IV), the smallest is \( -7 \) from Function I. So the function with the smallest minimum value is Function I.

Answer:

The function with the smallest minimum value is the first one (Function I, represented as "I" in the options).