How can you write a quadratic function in factored form …

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

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.

Short Answer

The answer outlines the steps to find the roots of a quadratic function given a vertex at x = 3. It explains that the roots can be expressed as (3 + r) and (3 – r), where r is a non-zero distance from the vertex, and concludes by showing how to write the quadratic in its factored form using these roots.

Step-by-Step Solution

Step 1: Understand the Vertex and Roots

In a quadratic function, the vertex provides essential information about the roots. In this case, we want the vertex to have an x-coordinate of 3, which helps determine the roots. The formula ensures that the x-coordinates of the roots can be expressed as (3 + r) and (3 – r), where r is a non-zero value representing the distance from the vertex to each root.

Step 2: Establish the Midpoint Property

The vertex’s x-coordinate acts as the midpoint between the two distinct roots. This means that the roots are symmetrically placed around the vertex. You can derive the roots using the following relationships:

  • Right Root: 3 + r
  • Left Root: 3 – r
As long as r is not equal to 0, these roots will be distinct.

Step 3: Write the Factored Form

To write the quadratic in its factored form, apply the roots into the standard template. The general equation of a quadratic based on its roots can be styled as follows:

  • Factor for Right Root: (x – (3 + r))
  • Factor for Left Root: (x – (3 – r))
Putting it all together, we can express the quadratic as a(x – (3 + r))(x – (3 – r)), where a is a scaling factor that adjusts the parabola’s width.

Related Concepts

Vertex

The highest or lowest point of a quadratic function, which indicates the maximum or minimum value and the x-coordinate helps in finding the roots

Roots

The solutions to a quadratic equation, represented as the x-values where the parabola intersects the x-axis, typically calculated using the vertex and a symmetric distance

Factored Form

A way of expressing a quadratic function that highlights its roots, formatted as a(x – root1)(x – root2), which allows easy identification of the roots and the shape of the parabola.

Table Of Contents
  1. 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.

Related Articles

Scroll to Top