Explain how you could write a quadratic function in factored form that would have a vertex with an x-coordinate of 3 and two distinct roots.
To write a quadratic function in factored form with a vertex of x-coordinate 3 and two distinct roots, we need to understand the concept of quadratic equations and how they can be expressed in factored form
To write a quadratic function in factored form with a vertex of x-coordinate 3 and two distinct roots, we need to understand the concept of quadratic equations and how they can be expressed in factored form.
A quadratic function is a second-degree polynomial function of the form: f(x) = ax^2 + bx + c, where a, b, and c are constants. The x-coordinate of the vertex of a quadratic function is given by the formula: x = -b / (2a).
In factored form, a quadratic function is expressed as the product of its linear factors. For a quadratic function with two distinct roots, the factored form is given by: f(x) = a(x – r1)(x – r2), where r1 and r2 are the roots of the equation.
To find our quadratic function with a vertex at x = 3, we can use the following steps:
Step 1: Determine the vertex form of the quadratic function.
Since the vertex of our function is at x = 3, we substitute 3 into the formula for the x-coordinate of the vertex: 3 = -b / (2a). This equation can be rearranged to give: b = -6a.
Step 2: Find the roots of the equation.
Since we require two distinct roots, we can assume that the roots are at x = 3 + h and x = 3 – h, where h is a unique constant. For simplicity, let’s assume h = 1.
Step 3: Substitute the values of the roots into the factored form of the quadratic function.
With the roots x = 3 + 1 and x = 3 – 1, we can rewrite the factored form: f(x) = a(x – (3 + 1))(x – (3 – 1)).
Simplifying this expression, we get: f(x) = a(x – 4)(x – 2).
So, the quadratic function in factored form with a vertex at x = 3 and two distinct roots is: f(x) = a(x – 4)(x – 2). The constant ‘a’ can take any value, such as 1, -1, 2, etc., to define the specific behavior and shape of the quadratic function.
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