To determine which graph represents the function \( f(x) = (x - 3)^2 \), we start by analyzing the characteristics of this quadratic function.
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Vertex: The vertex of the function \( f(x) = (x - 3)^2 \) is at the point \( (3, 0) \).
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Direction: The parabola opens upwards since the coefficient of the squared term is positive.
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Intercepts: The y-intercept can be found by substituting \( x = 0 \): \[ f(0) = (0 - 3)^2 = 9 \text{.} \] So the function passes through the point \( (0, 9) \).
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Symmetry: The parabola is symmetric around the vertical line \( x = 3 \).
Now, we can evaluate the given options based on these characteristics:
- First graph: Vertex at \((-3, 0)\), doesn’t match.
- Second graph: Vertex at \((0, -9)\), doesn’t match.
- Third graph: Vertex at \((3, 0)\) and passes through points \( (1, 4) \) and \( (5, 4) \), seems like it could match.
- Fourth graph: Vertex at \((0, 3)\), doesn’t match.
Now, let's verify the points for the third graph:
- The vertex is \((3, 0)\), which is correct.
- To see if it goes through \((1, 4)\): \[ f(1) = (1 - 3)^2 = (-2)^2 = 4 \text{, which is correct.} \]
- The point \((5, 4)\): \[ f(5) = (5 - 3)^2 = (2)^2 = 4 \text{, which is also correct.} \]
Since the third graph correctly fits all criteria:
The correct graph representing the function \( f(x) = (x - 3)^2 \) is the third graph, which has the vertex at \( (3, 0) \) and passes through \( (1, 4) \) and \( (5, 4) \).
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