# Vertex Form of a Line? Really?

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I read Glen’s post Writing Linear Equations about a month ago.  I remember thinking…  hmmm. really?  but a line doesn’t have a vertex, does it? is that legal?  can I do that?  Will it always work?  So I tried some.  Several.  It made perfect sense.

Anyway, I reread within the week and wondered if I should give it a try – I mean, it did make sense.  A lot of sense.  But I was afraid it would confuse the students, I was afraid to step out on the limb.  I wasn’t sure how to introduce it.

After seeing a tweet from @druinok, I decided to give it a try.  I shared the form y=a(x-h)+k and asked the students if they had ever seen it before?  In the functions unit – transforming functions (focus on quadratics, absolute values and radicals/square roots) – they were familiar.  I graphed a line with a given slope, then translated it (a point marked at then origin) the given point (h, k) –  to begin as a visual.

Anyway, within 3 examples – EVERY single student was doing it – writing equations of lines…CORRECTLY!  yes.  Thank you Glenn & Druinok!

In my geometry classes, I still had a handful of students struggling with this “algebra 1” skill.  I pulled them off one-on-one and simply said, lets try it this way, explaining the vertex form.  They are all running with writing equations of lines now~  AWESOME!  I am a believer in vertex form of a line.  Give it a try.  Go ahead don’t be afraid.  It works, beautifully.

### 5 responses »

1. I liked the idea of this as well but was equally put off by using the word ‘vertex’ in this context. However, I realized it was a form of what I usually teach as Point-Slope Form, and it gave me a connection I don’t recall previously making between quadratics and function shifts and lines. I’ll be using it in class soon!

2. Thank you Pam for the shout out! Every teacher I have spoken to who has tried this has seen their learners improve on lines. It is such a simple idea, but it makes sense in learners heads.

Aaron, I agree, lines don’t have vertices, but the form for quadratics, absolute value, etc is the vertex form. I think a better name, as I have been thinking about the class of general equations is the (h,k) form. That name is descriptive, yet also fits for lines as well as quadratics, abs value, etc.

• in the world where such things matter (which may or may not be this one), (h, k) is a “special point” … any specific point for linear, vertex of a quadratic, point of inflection for a cubic, I believe a ‘cusp’ might be the correct word for absolute value and center for a rational function, and perhaps ‘initial point’ for a square root (though it is the vertex here within the context of a restricted inverse function) (and so on and so on)? I love the idea of better connecting functions and transformations via this idea … point-slope form does not often seem to be the method of choice in a traditional Algebra 1 textbook; I would assume because using slope-intercept form emphasizes the key skills of previous chapters (specifically substitute/evaluate and solving equations). I had previously never thought of manipulating the variables in Point-Slope Form of a Line $y - y_{1}= m (x- x_{1})$ to create something that looks more like Vertex Form of a Quadratic, so for that I applaud you, sir.

3. Thank you Aaron, for the applause. I agree that nomenclature is important and for teachers to feel comfortable with using a different form it must have a name that fits with the rest of the established mathematics.

I am writing this up for submission to more authoritative sources, and clearly for that there needs to be a name that makes sense. I just am not sure what that name should be.

Or…. here is a thought. These equations can be called the (h, k) class of equations, and what makes the equation with degree 1 special in this category is that there is not one special point, but all points can act as (h,k). If you have any exponent other than 1 (or absolute value) then only one special point can be used.

This difference is what makes lines unique and special (and simpler) in the realm of mathematics, but also why there are other forms that work so well.

Hmm. Not sure if that makes sense or not, but it is an explanation that fits in my head. I would love some feedback!