Quadratic Functions: Vertex Form
Lesson 6 of 16
Objective: SWBAT identify algebraic forms of quadratic functions and create accurate graphs of quadratic functions presented in vertex form.
As a warm-up, I present students with a quadratic expression in vertex form and ask them to use the template Warm-Up Quadratic Functions to practice translating the quadratic into its other algebraic forms, graphing, and reporting interesting features of the graph. It is important to choose a quadratic that has integer x-intercepts and vertex coordinates (like y=2(x+1)^2-8), so I select the expression carefully. I make a second copy of this template on the back of the page so that students who complete the exercise quickly can be given a second, more challenging quadratic function to work with. I find that students generally enjoy the puzzle-like aspect of this activity.
As students complete the warm-up, I check homework with the homework rubric and show Desmos Quadratic in Vertex Form on the overhead projector with the "h" animation turned on. This will show the graph of a quadratic function that has the x-coordinate of the vertex changing gradually from -5 to 5 and back again. Watching this animation helps students understand the role of the parameter "h" in changing the location of a parabola in the coordinate plane. If students do not ask about this animation, I will initiate a discussion about what is changing in the graph displayed on the overhead and how it is related to the algebraic representation of the function [MP7].
If there are questions on the homework I will spend a few minutes answering them before we begin our discussion of standard and intercept form of quadratics. As always, I try to minimize time spend going over homework by providing answers on Edmodo and asking students to check their answers before they arrive in class.
As we begin to move forward with this lesson, I summarize the previous day's discussion of quadratic functions (standard and intercept forms). I also remind students that they should know a third quadratic form that is, in many ways, the most useful form. I continue the metaphor of algebraic forms being like "outfits" that the quadratic can wear. I then say, "Like slope-intercept form for a line, vertex form is the most useful for graphing a quadratic."
We dive right in and we discuss steps for graphing. We also practice a few graphs while taking notes on the algebraic manipulations required to translate from one quadratic form to the other [MP7]. I keep students involved in the discussion by asking for advice on how to proceed at key steps. Students will generally offer that translating TO standard form involves distributing and collecting like terms and that translating from standard to intercept form involves factoring, but they are stumped when it comes to putting a quadratic function into vertex form. The reflection "Creating a Need to Know, discusses how I capitalize on the gap in students' knowledge in order to engage students in direct instruction about completing the square.
At this point I introduce students to the Completing the Square Method of rewriting a quadratic function into vertex form. We examine the structure of quadratic functions in vertex form, noticing that there is a squared binomial in the form. I pass out Completing the Square Worksheet 1 and give students 5-10 minutes to practice writing perfect square trinomials as squares of binomials. My goal is to help students gain fluency with one part of a process which can be difficult. Becoming fluent in this step requires students to notice the regularity in the structure of perfect square trinomials [MP8].
We return to the discussion of completing the square and I show students a step-by-step process for completing the square to write a quadratic function in vertex form.
Practice and Reinforce
Students generally find the process of completing the square challenging and will need practice with this skill before they can use it flexibly in the context of graphing and interpreting quadratic functions. To improve fluency, students work on Completing the Square Worksheet 2, a collection of 10 quadratic functions in standard form that students translate into vertex form by completing the square. [MP7] I request that students check their work as they go by graphing both representations of each function on the graphing calculator to make sure they coincide [MP5].
As students work, I use the 3-Cup System to manage student frustration levels while providing a challenging experience for all students [MP1]. There may be times that I bring the whole group together to discuss something that is difficult for many students, but for the most part I prefer to answer questions at individual tables.
Exit Ticket and Assignment
After 40 minutes of practice, I bring students together for a discussion of the day's work. I use Complete the Square Quick Polls to determine if students are able to translate from standard to vertex form by completing the square.
I pass out the problem set Exploring Quadratics Vertex Form, which will be the evening's homework. Students are given time to look over the 4 parts of this assignment and ask clarifying questions. I remind students that solutions to this set will be available on Edmodo and that they should check their work before arriving in class the following day.