Previously, students used completing the square to rewrite quadratic expressions as perfect squares so they could solve equations. They also completed the square to derive the quadratic formula, which makes it possible to solve any quadratic equation. In this lesson, students encounter another use for completing the square—it can be used to rewrite a quadratic expression from standard form to vertex form.
Students explored the vertex form in a previous unit on quadratic functions. The lesson begins with a review of the form, its advantage, and its connections to the graph. Then, students recall how to transform expressions in vertex form into standard form, and then experiment with transforming the same expressions back to vertex form. Students notice that to take an expression from standard form to vertex form is essentially to complete the square, while being careful not to change the value of the expression.
To transform expressions into vertex form, students need to look for and make use of structure (MP7).
- Analyze and explain (orally and in writing) the steps for completing the square and understand how they transform a quadratic expression from standard to vertex form.
- Identify the vertex of a graph of a quadratic function when the expression that defines it is in vertex form.
- Write equivalent quadratic expressions in vertex form by completing the square.
- Let’s see what else completing the square can help us do.
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)
- I can identify the vertex of the graph of a quadratic function when the expression that defines it is written in vertex form.
- I know the meaning of the term “vertex form” and can recognize examples of quadratic expressions written in this form.
- When given a quadratic expression in standard form, I can rewrite it in vertex form.
vertex form (of a quadratic expression)
The vertex form of a quadratic expression in \(x\) is \(a(x-h)^2 + k\), where \(a\), \(h\), and \(k\) are constants, and \(a\) is not 0.