The Axis of Symmetry and Vertex Form

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SWBAT understand why the formula for the axis of symmetry is what it is, and to gain some experience using the vertex form of a quadratic expression.

Big Idea

As teachers, it's so important that we pay attention to what our kids can actually do, and make small adjustments accordingly. Today is a day to practice that!

About This Lesson

The exact shape of this lesson is different for every class I teach.  Over the course of the last four class meetings, I've tried to push kids as far as I can to practice graphing quadratic functions and to notice as much structure and as many connections as possible.

I'd like to take this lesson to lay out a few possibilities for what might happen today, but also to encourage you to think about the ways that you might adapt an "intermediate day" like this one to any given class.

The one goal I have with all sections is to provide the formula for the axis of symmetry.  How I do so depends on the class.  Maybe kids have all sorts of connections that they can already make, maybe they won't be surprised by this formula, maybe they'll be mad at me at that I didn't just share this sooner, or maybe they'll be relieved to have this convenient tool.

If you've used any or all of the previous four lessons, you already might have great ideas for what your kids need to do today.  Run with those ideas!  Think of each section of this lesson as a snapshot of what might happen today.

Opener: Find these Roots

5 minutes

Today's opener is on the first slide of the lesson notes, and it prompts students to find the roots of three similar quadratic functions.  I want students to notice that the middle coefficient is the same for each of these three functions, and to develop their own conjectures (maybe they've already been doing this on their own!) about how that similarity might express itself on the graphs of each of these functions.

By now, I hope to see my students gaining confidence in finding roots by factoring.  I allow everyone a few minutes to do this on their own, and I circulate to check in with each table.  If I see that a student is struggling, I'll pair them up with someone who is good at it - often, these students are already sharing a table - and ask them to discuss the task.  It's great to identify student experts and to allow them to share that expertise.  Usually, kids are better teachers than I could ever be.

After a few minutes, I elicit help from the class, and we record all solutions on the board.

Guided Notes: What Do You Notice?

5 minutes

Now that we've got the roots for each of these functions, we can review what we saw two days ago: that the axis of symmetry and the vertex are halfway in-between the two roots.  The second slide of the lesson notes prompts students to find the axis of symmetry and the vertex for each of the three functions in the opener.  Of course, when we do this, we see - what do you know? - that the axis of symmetry is the same for each.

I circulate as students get to work, and then I set up a chart on the side board, that we can use to keep track of what we find.  As we get to filling in the first three rows of the chart, it's clear what these three functions share.  At this point, kids might start to suspect that there's indeed a relationship between the middle coefficient and the axis of symmetry.  If they start to talk about that, I might wonder aloud about the relationship between 8 and -4.

The third slide prompts us to repeat the process of finding the roots, the axis of symmetry and the vertex for three more quadratic functions, and you'll see that I snapped the accompanying photo of the chart midway through that work.  The purpose of these three functions is to give students further evidence that the axis of symmetry is closely related to the middle coefficient, and hopefully anyone with a conjecture will be able to make some predictions here.

The fourth slide asks, What do d, e, and f have in common?  At first glance it might be hard to tell, but upon completing the task it should be clear that the vertex of each has a y-coordinate of -9.  This provides a springboard into thinking about vertex form, which makes that connection even more clear.  When we get to that (as slides 5 and 6 prompt us to do), I might ask, "What do I have to add to each of these functions in order to complete the square?"

From there, a discussion might start about how the vertex appears in the vertex form representation of each function.  Note that I've used the same modified version of vertex form - with a plus sign inside the parentheses - that I used on the Quadratic Functions in Three Forms activity.  Students will see the traditional form

f(x) = (x-h)^2 + k

with its vertex at (h,k), when it's their idea to make that adjustment.

The Formula for the Axis of Symmetry

5 minutes

As I note above in the "About this Lesson" section, the one objective I insist on getting today is to share with students the formula for the axis of symmetry.  Namely, that for any quadratic function in standard form

f(x) = ax^2 + bx + c

where a does not equal 0, the axis of symmetry is given by 

x = -b/(2a)

The seventh slide of the lesson notes provides some blank space to write that definition on the board.  

There are no frills here.  I keep this part as straightforward as possible, and I want to provide space for kids to say, "I knew it!"  In my teaching, I find it most satisfying to share a formula like this and to see that kids already have the intuition that this is true.

To apply the formula, I show the class that we can quickly find the vertex of a function by using this formula - often making the calculation in our heads - before getting the y-coordinate by evaluating the function for that value of x.

Options for Practice

23 minutes

With any remaining time, students have a few work options today.  The remaining slides are to be used as needed, depending on the progress on any class.  Whatever we do will involve spiraling back to some of the work that students have seen in preceding days.

Slide #9 is re-used from Lesson 10: A Faster Way to Graph a Parabola, and can be used when most students still have some work to do on that.  Knowing what they know now about the axis of symmetry should provide the confidence boost necessary to get this done.

Slide #10 lists two more options.  One is for students to take another look at the Gallery Walk from earlier in the week.  Eleven functions are still posted around the room, and this time, students are instructed to rewrite each function in vertex form and to sketch a quick graph of each function.  It's a different task from the original assignment, and I tell students that anyone who wants to re-submit this assignment can do so.

The other option is to return, once again, to the Quadratic Functions in Three Forms assignment.  Many students had fine success in completing the first two columns of that chart, but left the third column - vertex form - blank.  After today's lesson, many of those kids are prepared to try again as they practice rewriting quadratic expressions in vertex form.

The last four slides list the green and blue functions from the gallery walk, which gives me the ability to review these sets of functions in a whole-class or small group discussion.