Identifying the Graph of the Quadratic Function f(x) = (x – 3)^(2): A Comprehensive Analysis and Comparison of Options

Which graph represents the function f(x) = (x – 3)2?

To determine which graph represents the function f(x) = (x – 3)^(2), we can first identify the key characteristics of the function and then examine the options provided

To determine which graph represents the function f(x) = (x – 3)^(2), we can first identify the key characteristics of the function and then examine the options provided.

The function f(x) = (x – 3)^(2) represents a quadratic function in the form of f(x) = a(x – h)^(2) + k, where (h,k) represents the vertex of the parabola. In the given function, (3,0) is the vertex of the parabola.

Key characteristics of the function:
1. Vertex: The vertex is at (3,0).
2. Axis of Symmetry: The axis of symmetry is a vertical line passing through the vertex, which is x = 3.
3. Opening direction: Since the coefficient of (x – 3)^(2) is positive (in this case 1), the parabola opens upward.
4. Symmetry: The graph is symmetric with respect to the axis of symmetry.

Now, let’s examine the options provided and see which graph matches the characteristics of the given function:

Option A:
This graph does not have its vertex at (3,0). It is not symmetric with respect to the axis of symmetry x = 3. Therefore, option A does not represent the function f(x) = (x – 3)^(2).

Option B:
This graph has its vertex at (3,0), which matches the given function. Furthermore, it is symmetric with respect to the axis of symmetry x = 3. Therefore, option B represents the function f(x) = (x – 3)^(2).

Option C:
This graph appears to have its vertex at (-3,0), which is not consistent with the given function. It is not symmetric with respect to the axis of symmetry x = 3. Therefore, option C does not represent the function f(x) = (x – 3)^(2).

Option D:
This graph also appears to have its vertex at (-3,0), which is not consistent with the given function. It is not symmetric with respect to the axis of symmetry x = 3. Therefore, option D does not represent the function f(x) = (x – 3)^(2).

Based on our analysis, the graph represented by option B matches the characteristics of the function f(x) = (x – 3)^(2).

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