To write a quadratic function in factored form with a given vertex and distinct roots, we can use the properties of quadratics and how to express them in a specific way.
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Understanding the Vertex Form: A quadratic function can be expressed in vertex form as: \[ f(x) = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola. Since we want the vertex to have an x-coordinate of 3, we can establish that \(h = 3\).
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Choosing k: To find a specific function, we need a value for \(k\). For this example, let's choose \(k\) to be a value that allows the function to have distinct roots. We can set \(k = 0\) for simplicity, but any value that does not yield a double root will work.
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Referring to the Factored Form: The factored form of a quadratic function with roots \(r_1\) and \(r_2\) is: \[ f(x) = a(x - r_1)(x - r_2) \] We need to ensure that the roots \(r_1\) and \(r_2\) are real and distinct, which means we need to have \(r_1 \neq r_2\).
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Finding the Roots: We know that the vertex \( (3, k) \) will be the midpoint between the two roots \( r_1 \) and \( r_2 \). Thus, we can express the roots in terms of the vertex: \[ r_1 = 3 - d \quad \text{and} \quad r_2 = 3 + d \] where \(d\) is some positive number (to ensure distinct roots).
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Writing the Factored Form: Plugging \(r_1\) and \(r_2\) back into the factored form, we get: \[ f(x) = a(x - (3 - d))(x - (3 + d)) = a(x - 3 + d)(x - 3 - d) \]
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Choosing a Value for a: The value of \(a\) can be any non-zero number, and it determines the direction and width of the parabola. For example, we can simply choose \(a = 1\).
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Final Function: If we choose \(d = 1\), then the roots become \(r_1 = 2\) and \(r_2 = 4\). The function in factored form with \(a = 1\) would be: \[ f(x) = 1(x - 2)(x - 4) \] Thus, the function is: \[ f(x) = (x - 2)(x - 4) \] This function has roots at \(x = 2\) and \(x = 4\), and the vertex (the midpoint of the roots) is at \(x = 3\).
In conclusion, a quadratic function in factored form with a vertex at \( x = 3 \) and two distinct roots can be constructed using specific choices for \(a\) and the distance \(d\) from the vertex to the roots.