## Tuesday, November 18, 2014

### Square and Cube Root Graphs with Geogebra

Yep, it's that time again.  We are back to graphing, and we are back to transformations.  The book that I use for Algebra II starts in Chapter 2 with transformations of the absolute value function. Chapter 4 includes quadratics, Chapter 5 is cubics, and now in chapter 6 it is square root and cube root functions.  We began the lesson by looking back at the Desmos pre-made sheet for parabolas in vertex form for a reminder of how a, h, and k transform a graph.

I decided to take a different approach this time around for the rest of the lesson.  Instead of starting with functions and table of values, we started with a transformation given in words.

I showed the Geogebra applet below.  Directions are to move points A, B, and C so they are shifted up from the parent graph by two.  Students could pick which of the points from the parent graph to look at, as long as A, B, and C are two units up from a point on the parent graph of y=sqrt(x).

Next I typed the y=a*sqrt(x-h)+k into the input bar, and we talked through the transformations that had taken place.  I changed a, h, and k as we decided on values.  For this example a=1, h=0 and k=2.

If you have entered the correct function into the input bar (hit enter when finished), the transformed function will turn green if correct.

Students practiced a few problems with both horizontal and vertical translations, and then problem 5 asked students to reflect y=sqrt(x) about the x-axis.  About 90% of students were able to move points A, B, and C to the correct location, and about 50% correctly guessed that the correct transformed function was y=-sqrt(x).  This was with NO direct instruction for this part of the lesson.  I was hoping for this moment, as one of my goals for this approach was for students to think about how we were transforming a set of points, and specifically to think about what happens to a set of points before we think about the transformed function.

I modeled one more problem for the class, and then students spent the rest of class working through this problem set.  There is space to record transformed points and the transformed function.  Answers can be checked using the applet, which is set up to indicate a correct answer as long as your transformed function went through points A, B, and C.  Since students were checking their own answers, I got to walk around and asked questions to check for understanding (love this!).  I made a standard homework assignment (graph the given function) so that students would also practice problems that are similar to those that will be on the quiz/test.  I'm definitely interested to see how well they did on the HW, since it is not the same as the classwork.  Overall, I like this approach for introduction to graphing a new type of function in Algebra II, and I can see adapting this lesson for the other function types (with tweeks of course!).

The Geogebra Applet used in this lesson is available HERE.  One last note is that it is possible for students to get the wrong "answer" using this applet.  The floating points A, B, and C begin at (1,4), (4,4) and (7,4), so if a student enters the function f(x)=4 the applet will indicate that they are correct.  This has happened a few times since I started using applets like this one, and it's made for great conversation and reinforces the need to check the reasonableness of an answer.