The vertex of a quadratic function in factored form, f(x) = a(x - r₁)(x - r₂), is found by first determining the x-coordinate of the vertex (axis of symmetry) as the midpoint between the roots r₁ and r₂, and then substituting this x-value back into the function to find the y-coordinate.
Here's a step-by-step breakdown:
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Identify the Roots (x-intercepts): The factored form immediately gives you the roots of the quadratic equation. These are the values of x that make f(x) = 0. The roots are r₁ and r₂, found by setting each factor (x - r₁) and (x - r₂) to zero and solving for x.
- x - r₁ = 0 => x = r₁
- x - r₂ = 0 => x = r₂
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Calculate the x-coordinate of the Vertex (Axis of Symmetry): The x-coordinate of the vertex lies exactly in the middle of the two roots due to the parabola's symmetry. Calculate it using the midpoint formula:
- x_vertex = (r₁ + r₂) / 2
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Calculate the y-coordinate of the Vertex: Substitute the x-coordinate of the vertex (x_vertex) back into the original factored form of the quadratic function to find the corresponding y-coordinate (y_vertex):
- y_vertex = a(x_vertex - r₁)(x_vertex - r₂)
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Write the Vertex: The vertex is the point (x_vertex, y_vertex).
Example:
Consider the quadratic function f(x) = 2(x - 1)(x - 5).
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Roots: The roots are r₁ = 1 and r₂ = 5.
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x-coordinate of Vertex: x_vertex = (1 + 5) / 2 = 3
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y-coordinate of Vertex: y_vertex = 2(3 - 1)(3 - 5) = 2(2)(-2) = -8
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Vertex: The vertex is (3, -8).
In summary, finding the vertex from the factored form relies on the symmetry of the parabola. By averaging the roots, you find the x-value of the vertex, and then plugging this value back into the function provides the y-value.
