## Sunday, December 28, 2014

### Practice your inverses with Geogebra

Geogebra is great for any concept that involves graphing, and inverse functions are no exception.  I was planning for this topic recently in Algebra II, and saw that my textbook included the following 7 examples for this section:

1. Find an equation of an inverse function.
2. Verify that two functions are inverses of each other.
3. A multi-step word problem
4. Find an inverse and graph both functions.  Restrict the domain.
5. Graph a function, use horizontal line test to decide if inverse exists, then find inverse.
6. Find inverse of a power model (so y=a*x^b)
7. Use the previous inverse power model to make a prediction.

Ouch!  That's a lot cover in one section of Algebra II.  A lot of the conceptual understanding of inverse functions involves understanding the relationship between the algebraic and graphical representation of inverse functions.  It's not helpful in this section for students to spend a lot of time graphing by hand.  Graphing on Geogebra or Desmos as part of the day 1 intro will be helpful.  I created a day 2 or review type of activity to give students a chance to practice with putting some of the concepts together.  Paper worksheet is here, and Geogebra Worksheet is here.  Some directions for how to introduce the activity to students are on page 1 of the document, and also below.

1.  Open the Geogebra worksheet.

2.  Type function f into the input box, and hit enter.  For example, f(x)=(x-5)/3.  Move points A, B, C to find the three points on function f.  They will turn blue when you move them on the function.  Record these points on your paper for f(x)

3.  Switch x & y to get points on the inverse function.  Record these points on your paper for f-inverse, and move A, B, and C to the points that should be on the inverse.

4. Find f-inverse by showing the algebraic steps on your paper.  Then type your answer for f-inverse into the input box on Geogebra and press enter.  If you have the correct answer you will see that f-inverse goes through points A, B, and C, and it will turn green to let you know that you are correct.

Easy to use and self checking.  Just reload the page when you are finished to try another example.

Geogebra can be picky when it comes to notation.  We noticed two types of functions to be aware of.
1.  When typing in a cube root (or odd root) function, be sure to include a coefficient, even if it is just 1.  So for f(x)=x^(1/3), type f(x)=1*x^(1/3).
2.  For a log function with base 2, type f(x)=log(2,x), or in general type f(x)=log(a,x) for base a.  The default base is e.

If you like this type of self-checking Geogebra activity, check back soon to see more.  I'll be posting soon for transformations of quadratics, as well as graphing cubics and quartics in intercept form.  I also posted a similar activity in November on graphing square and cube root functions.