This resource is meant to supplement lessons on completing the square (quadratic equations) in order to find the vertex of a quadratic function and write the function in vertex form. The practice problems provide scaffolding to allow students to slowly build up to fully using the process of completing the square.

Students will be able to rewrite quadratic expressions given in standard form, ax^2 + bx +c (with a not equal to 1), as equivalent expressions in completed-square form, a(x - h)^2 + k. They build quadratic expressions in basic business application contexts and rewrite them in equivalent forms.

Unit with explanations, definitions and practice on completing the square for purposes of defining max and min.

In this Khan Academy activity, students will determine the zeros and vertex of a quadratic equation.

This video rewrites the equation y=-5x^2-20x+15 in vertex form (by completing the square) in order to identify the vertex of the corresponding parabola.

This lesson unit is intended to help teachers assess how well students are able to understand what the different algebraic forms of a quadratic function reveal about the properties of its graphical representation. In particular, the lesson will help teachers identify and help students who have the following difficulties: understanding how the factored form of the function can identify a graphŐs roots; understanding how the completed square form of the function can identify a graphŐs maximum or minimum point; and understanding how the standard form of the function can identify a graphŐs intercept.

In this Khan Academy activity, students will graph a parabola given any form,

In this Khan Academy activity, students will graph a parabola given its standard form.

The purpose of this task is to build fluency in writing equivalent expressions for quadratic equations using factoring, completing the square, and the distributive property. Students will use the equations that they have constructed to analyze and graph quadratic functions.

The purpose of this task is two-fold. The first purpose is for students to explore and generalize how the features of the equation can be used to graph the quadratic function. The second purpose is for students to deepen their understanding of quadratic functions as the product of two linear factors. In the task, students are asked to graph parabolas from equations in factored form. They are given several cases to provide an opportunity to notice how the x-intercepts, yintercept, and vertical stretch are readily visible in the equation. This also sets them up to notice the relationship between the ! -intercepts and the y-intercept. The task extends this thinking by asking students to start with any two linear functions, multiply them together and find the function that is created, which is quadratic. They graph both the initial lines and the parabola to find the relationship between x-intercepts and y-intercept and to highlight the idea that quadratic functions are the product of two linear factors.

Image of parabola on coordinate plane, with hot spots indicating focus and directrix, along with an example of deriving the equation for a parabola given the focus and directrix.

This lesson examines the process of completing the square as a method of solving quadratic equations.

This video shows an example of how to solve a quadratic using completing the square.

This task provides opportunity to extend the work of factoring and working with the area model for quadratics to those of the form !!! + !" + !, with an a-value other than one. Students will work to see how the area model connects with quadratics of this form and how both factored form and standard form connect with the area model. The task begins with expressions that have a common factor between terms, and continues to other expressions with ! ≠ 0. The distributive property will be used to verify the work and move to efficiency as combinations of the factors of a are considered with the combinations of the factors of c.

The purpose of this task is for students to understand equivalent expressions obtained from factoring trinomials. In the task, students use area model diagrams to identify the sides of the rectangle, and thus, the factors. In the previous task, Factor Fixin’, students factored trinomials in which all of the terms are positive. This task builds on that work to include factoring expressions that have both positive and negative terms. The problems are carefully selected to help students see number patterns that they can use to become fluent with factoring. Students write expressions in both factored form and standard form.