To determine which graph models the function [tex]\( f(x) = 2(x-4)^2 + 3 \)[/tex], we need to analyze the key features of the function:
1. Vertex Form of the Parabola:
The given function [tex]\( f(x) = 2(x-4)^2 + 3 \)[/tex] is already in vertex form [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola and [tex]\( a \)[/tex] determines the direction and width of the parabola.
2. Identify the Vertex:
From the function [tex]\( f(x) = 2(x-4)^2 + 3 \)[/tex]:
- Compare with [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where [tex]\( a = 2 \)[/tex], [tex]\( h = 4 \)[/tex], and [tex]\( k = 3 \)[/tex].
- Therefore, the vertex of the parabola is [tex]\( (4, 3) \)[/tex].
3. Direction of the Parabola:
- The coefficient [tex]\( a \)[/tex] (in this case, [tex]\( a = 2 \)[/tex]) is positive, meaning the parabola opens upward.
4. Summary of Key Features:
- Vertex: [tex]\( (4, 3) \)[/tex]
- Direction: Upward
Based on these features, you should look for a graph with the following characteristics:
- The vertex is located at the point [tex]\( (4, 3) \)[/tex].
- The parabola opens in an upward direction.
Graphs labeled A and B can now be compared against these features to identify the correct one. Look for the graph that accurately represents these distinctive features of the function [tex]\( f(x) = 2(x-4)^2 + 3 \)[/tex].
Determine which graph has the vertex at [tex]\( (4, 3) \)[/tex] and opens upwards. That graph will be the one that correctly models the function [tex]\( f(x) = 2(x-4)^2 + 3 \)[/tex].
The correct graph is the one matching these criteria.
