Understanding the Relationship Between Vertex and Roots | Writing a Quadratic Function in Factored Form

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 at an x-coordinate of 3 and two distinct roots, we need to understand the relationship between the vertex and the roots of a quadratic function

To write a quadratic function in factored form with a vertex at an x-coordinate of 3 and two distinct roots, we need to understand the relationship between the vertex and the roots of a quadratic function.

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

In this case, we know that the x-coordinate of the vertex is 3, so h = 3.

Let’s consider the roots of the quadratic function. Since there are two distinct roots, it means that the parabola intersects the x-axis at two different points.

In factored form, the quadratic equation can be written as: f(x) = a(x – r1)(x – r2), where r1 and r2 are the roots of the equation.

Since the quadratic has two distinct roots, we can express it utilizing the factors (x – r1) and (x – r2), where r1 and r2 are the two roots.

However, to find r1 and r2, we need to consider what happens to the quadratic equation at the vertex. We know that the x-coordinate of the vertex is 3, so substituting x = 3 into the equation gives us the y-coordinate of the vertex, which we’ll denote as k.

So, the y-coordinate of the vertex is f(3), which is equivalent to a(3 – h)^2 + k.

Since the vertex is on the parabola, it means that (3, k) satisfies the quadratic equation. Hence, substituting x = 3, we have:

k = a(3 – 3)^2 + k
0 = 0

From this equation, we deduce that k = 0.

Now, let’s consider the roots r1 and r2. Since the vertex occurs at x = 3, and we have two distinct roots, we can assume that those roots are symmetrically located around the vertex on either side.

Hence, r1 = 3 – d and r2 = 3 + d, where d represents the distance from the vertex to one of the roots.

Substituting these values into the factored form of the quadratic equation, we get:

f(x) = a(x – (3 – d))(x – (3 + d))
f(x) = a((x – 3 + d)(x – 3 – d))
f(x) = a((x – 3)^2 – d^2)

So, the quadratic function in factored form with a vertex at an x-coordinate of 3 and two distinct roots is:

f(x) = a((x – 3)^2 – d^2), where d is the distance between the vertex and one of the roots.

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