Transformation of Functions Day 1

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Objective

SWBAT determine a connection between the structure of a functions equation and the appearance of the graph of the function.

Big Idea

Students will use structure and patterns to determine the "rules" for translating functions.

Opening

3 minutes

This opening allows students to connect the topic of transformation of functions to a concept that they have already learned in previous grades.  Once you project transformations_open1, have students do a Quick Write regarding what happened to move from the first image to the second image.  Tell them to be as specific as possible when describing what happened.  Once students have a minute to jot down their thoughts have them turn-and-talk to describe the transformation and give a rationale for their description (MP3).

At this point I have a couple of students share out how they knew that the triangle was shifted 7 units to the right.  Most students will say that they tracked one of the points (either a, b, or c) to find out how much it moved.  Really focus on this idea because when the students investigate the movement of parabolas they will also be focusing on one point, the vertex of the parabola.

Launch

7 minutes

In the transformations_launch presentation, students will first see a slide which structures the process of describing linear transformations of functions. In the subsequent slides, students are given an original parabola (in blue) and asked to describe the transformation to the new parabola in red.

I have students do a Think-Pair-Share as they look at the first transformation.  As students share with their partners, I listen for those who are discussing how far the vertex moved.  This tells me that they are using the important idea of following one point on the graph

Note: Unless students have studied parabolas they may not use the word vertex.  Many students say "bottom point".  You can introduce the term during the share out so that students can use it in their explanations during the investigation

We go through the second transformation a little more quickly as students will now be catching on to the idea.  Point out to students that the parabola is not getting more narrow (many students will think this).  Show them that if you shifted the red parabola back down it would match up with the blue parabola.

Investigation

25 minutes

The transformations_investigation walks students through transformations of a parabola in all four directions on the coordinate plane.  Let students work through the investigation with a partner using a calculator to graph each function.

Note: I had students use a TI-84 graphing calculator.  Each student graphed y=x^2 in the y1 line and I had them bold this function.  This can be done by keying over to the far left of the equation and then hitting enter one time.  Here is a link to perform this function if you are not sure how.

Ensure that students are describing both the direction and number of units for each transformation. Students should notice relatively quickly that the constant that is being added either inside or outside of the parenthesis will be the amount of movement.  The next challenge is to determine a pattern for making the graph move up, down, left and right.  Students will need to look at the structure of each equation to make these generalizations (MP7).

Closure

5 minutes

For the closing activity students should write down all five functions on a half-sheet.  Once they write down all five functions, put up the next slide and have the students match each function with the appropriate graph.  After they match the graphs I also have the students write what the transformation was for each function (example: the graph shifted 8 units to the left).  I want to continue to have them describe the transformation to deepen the connection between the structure of the equation and the resulting transformation.