Question

drag each label to the correct location on the table. each label can be used more than once.
function\tkey features
x -2 -1 0 1 2 3
f(x) 5 0 -3 -4 -3 0\t
graph of g(x)\t
x -2 -1 0 1 2 3
m(x) -1 2 3 2 -1 -6\t
graph of f(x)\t

Explanation:

Step1: Analyze \( f(x) \) (table)

• Vertex: The minimum (since it's a parabola - shaped table, symmetric around \( x = 1 \) as \( f(0)=f(2)= - 3 \), \( f(-1)=f(3)=0 \)) at \( (1, - 4) \).
• Roots (x - intercepts): When \( f(x)=0 \), \( x=-1,3 \).
• Y - intercept: At \( x = 0 \), \( f(0)=-3 \).
• Symmetry: Axis of symmetry \( x=\frac{-1 + 3}{2}=1 \) (or from the table, symmetric around \( x = 1 \)).

Step2: Analyze \( g(x) \) (graph)

• Vertex: Maximum point (since it opens downward) at \( x = - 1 \) (mid - point of roots \( x=-3,1 \), so axis of symmetry \( x=\frac{-3 + 1}{2}=-1 \)).
• Roots (x - intercepts): \( x=-3,1 \).
• Y - intercept: Positive (crosses y - axis above origin).
• Symmetry: Axis of symmetry \( x=-1 \).

Step3: Analyze \( h(x) \) (table)

• Vertex: Maximum (since \( h(-2)=-1,h(-1)=2,h(0)=3,h(1)=2,h(2)=-1,h(3)=-6 \), symmetric around \( x = 0 \)) at \( (0,3) \).
• Roots (x - intercepts): None in the given \( x \) range (all \( h(x) \) values: - 1,2,3,2,-1,-6; no zero except maybe outside, but in table, no \( h(x)=0 \)).
• Y - intercept: At \( x = 0 \), \( h(0)=3 \).
• Symmetry: Axis of symmetry \( x = 0 \) (since \( h(-x)=h(x) \) for \( x=-2,-1,1,2 \)).

Step4: Analyze \( j(x) \) (graph)

• Vertex: Minimum point (opens upward) at \( x = 2 \) (mid - point of roots? Wait, roots are around \( x = 0.5 \) and \( x = 3 \)? Wait, graph shows it opens upward, vertex at \( x = 2 \), y - intercept positive, crosses x - axis at two points.
• Roots (x - intercepts): Two positive x - intercepts (around \( x = 0.5 \) and \( x = 3 \)).
• Y - intercept: Positive (crosses y - axis above origin).
• Symmetry: Axis of symmetry \( x = 2 \) (mid - point of roots).

(Assuming the task is to identify key features like vertex, roots, intercepts, symmetry for each function. Since the original problem says "Drag each label to the correct location", but labels are not shown. However, based on the analysis, we can match the key features to each function.)

Answer:

(The answer would depend on the labels provided, but based on the analysis above, for each function:

• \( f(x) \) (table): Vertex at \( (1,-4) \), roots \( x=-1,3 \), y - intercept \( - 3 \), symmetry \( x = 1 \).
• \( g(x) \) (graph): Vertex at \( (-1,y_{max}) \), roots \( x=-3,1 \), y - intercept positive, symmetry \( x=-1 \).
• \( h(x) \) (table): Vertex at \( (0,3) \), no roots in table, y - intercept \( 3 \), symmetry \( x = 0 \).
• \( j(x) \) (graph): Vertex at \( (2,y_{min}) \), two positive roots, y - intercept positive, symmetry \( x = 2 \).)

(Note: Since the labels are not provided in the question, the above is the analysis of key features for each function to help with the drag - and - drop task.)