# whY is X minus H & plus K?

We all have our methods of helping students gain an understanding of this:

Let’s start simple as most of us do in the classroom.  In years past, I (and many others) have helped students explore this concept with something like:

This concept develops as students explore more.  They learn that operations inside the primary piece of the function have horizontal effects, while outside correlates to vertical changes.  Easy enough.  But then, it happens.

Student: “Mr. Butler, I get that plus or minus moves the parabola.  But how come when you do vertical is goes the way it’s supposed to, and sideways is like opposite.”
Butler: “That’s a great question.  What do you mean by opposite.” (trying to get the student to articulate more specifically what he/she is trying to say while I ponder it myself in hoping to come up with an appropriate answer).
S: “Like vertical stuff goes up for plus, down for minus.  Just like the graph”
B: “What do you mean the graph?”
S: “The axis.  The y-axis.  But then for the other axis, the x-axis, it moves opposite.  Like negative is left, but when it’s minus it goes right.  And positive is right, but when it’s plus inside the parabola goes left.”
B: “Who else sees the pattern, or something like it?”

Lots of Hands.

B: “Well, you see whenever its inside, the transformation is opposite.  We’ll see more of that when we look at multiplying too.”

Cue the tilt head, “Huh?” OR “Oh …..I get it,” fake voice.

This process with examples didn’t answer the question conceptually, it just confirmed that there’s a pattern.  Yes, inside opposite, outside normal.  But why?

There’s an opportunity here to go deeper.  Something that I explored with students was the idea of changing inputs and outputs.  I’m pretty sure you’ve heard of the function machine:

Credit: Chris Robinson via 101qs.com

Students get this, kinda.  You choose an input, do some math, then get an output.  It’s helpful to reference the inside and outside as add-ons to the machine.

Students can kinda get a grasp on that, but it’s still abstract.  So we iterate with a few different function machines: quadratic, cubic, and maybe even some square root functions.

We often lean on words like shift and stretch, and then that whole confusion with the opposite comes in.  What is the opposite of stretch?  Do we say compress?  Can we compress by a negative?  What does that look like?  In this case many would favor more appropriate vocabulary like translate, reflect and dilate.  Either way through repetition and memorization we hope our students can eventually break down the effects of a,b,h and k for:

Wait, do we really understand what’s going on with the inside/opposite confusion yet?

## Finally, Some Insight with the Inside

When we are making a change to the inside, we are making a change to the input, before it goes into the function machine.  That’s like making a change to all the inputs at the same time.  We conveniently have an organized visual set of the inputs: the x-axis.  So any change to the inside/input is actually a change to the x-axis.  Play with this and see if it starts to conceptualize a little better for you now.

## The Applet

I’m not opposed to starting simple. In fact, I’m all for it.  There are so many layers in function transformation and it takes a while to handle what’s really going on under the hood.  I do suggest that once that question of “Why?” is primed and ready, get the students to explore with this.

courtesy of Samuel Winter via flickr CC-BY-NC-SA License
How would you graph?

Would you factor?  Would you complete the square?  Quadratic Formula?  Are those my only options?

The idea of multiple representations  of the same thing, or multiple methods toward the same end, has strong correlation for learning (and teaching) mathematics.  Jo Boaler demonstrates this in her number talks.  Dan Meyer and Robert Kaplinksy discuss these through the open middle problem design. One way we see this in polynomials (including quadratics) is…

# Factoring

There’s the famous XBox method, often abbreviated to a formulaic method by teachers.  I like this approach, especially because it incorporates area models, grouping, flexibility for multiple values of the lead coefficient “a”.    James Key (aka @iheartgeo), brings in more of the number talk by using powers of 10 in addition to powers of x
 .

Here’s a quick preview of his vine that relates:

Transitioning to more algebraic/abstract examples seems simpler when relating quadratics to numerical examples like this vine. He looks at the factors and the relationships between the terms.  Students in his class are prescribed a formula, but asked to make sense of the relationships and then transition to methods with more regularity.

# Complete the Square

A method often confusing to Algebra 1 students because of its abstract structure is again made simple with looping short videos.

Plenty of teachers try to support their process with visualization, all that is intended here is to start with the visual reasoning and pattern making, then move to abstract.

# The hybrid

It’s not exactly completing the square.  It’s not exactly factoring and using the roots.  It’s the compromise between the two.

• plot the points (0,0) and (-b,0)
• shift vertical c units.

Let’s make that easier.  Here’s a dynamic version:

# Open up the Middle

But wait, the two points (x1,0) and (x2,0) that plot on the horizontal axis can be any combination that adds up to your linear coefficient “b”.  The vertical shift just needs to balance those numbers with c.

So for our example: , the 8x needs to be split, and c needs to be balanced with the shift:

 x1 x2 c shift? 8 0 20 20 0 8 20 20 4 4 20 4 2 6 20 8 1 7 20 ? -2 10 20 ? 3 5 20 ?

Now with that bit of intuition, we can open up the middle and have the user do some exploration.  Here’s a question that encourages looking at multiple representations and seeing how the values are connected:

When thinking of factors, roots, and vertical shifts, the relationship becomes more dynamic.  We could revisit the open middle prompt and ask to define / express the relationship between a,b and c.   Struggling students can simplify the algebra using Jame’s Keys approach with substituting a value for x.  Here’s a Geogebra applet to help visualize the multiple representations:

Play with the applet and notice that as long as the quadratics share the same line of symmetry, only a vertical shift is needed to overlap.  This also reveals the congruence in some quadratics.  Ask yourself what makes the quadratics congruent?

Here’s a similar interactive graph built with Desmos.

# What now?

I would expect that this more flexible visualization allows for more depth in understanding how a quadratic function works, both visually and algebraically.  Students who are asked to match quadratic algebraic expressions may be little less intimidated by starting with some dynamic pictures.  So now let’s return to the original question (that has multiple possibilities):

How would you graph?

What other questions might you ask?