Question

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.

Answer 1

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.

1. **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 \(h = 3\).

2. **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.

3. **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 \(r_1 \neq r_2\).

4. **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).

5. **Writing the Factored Form**: Plugging \(r_1\) and \(r_2\) back into the factored form, we get:
\[
f(x) = a(x - (3 - d))(x - (3 + d)) = a(x - 3 + d)(x - 3 - d)
\]

6. **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 \(a = 1\).

7. **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.