Time allotment: 15 minutes
1. Direct students to the Math Is Fun Symmetry Artist.
2. Ask them to create their masterpiece. Give the students 3 or 4 minutes to play around and become familiar with the website.
3. Now instruct students to select the y=x reflection symmetry and tell them to create a quick drawing using this setting.
4. Have the students share a description of their drawings. (You want them to notice that the image is a reflection over the line.) You may choose to use an online bulletin board like Padlet to allow students to copy and paste a screenshot of their pictures along with a description of how the drawing was created. You are looking for a discussion of the reflection.
5. Discuss student descriptions, being sure to ask students to recall the method for reflecting over the line y=x.
Time allotment: 30 - 40 minutes
1. Explain to students that they will be finding the inverse of a function from its graph.
2. Show students the graph of y=x^3. Ask them to decide if the graph represents a function.
3. Remind students that to find an inverse you reverse the input and the output.
4. Have the students locate at least 3 points on the graph (you can lead them to choose (-2, -8), (0,0) and (2,8).
5. Next, ask them to use the points from step 4 to find points on the inverse and to sketch the inverse.
6. Ask students to imagine drawing the graph on the symmetry artist website. They should see that the sketch of the inverse would be the same drawing.
7. Now ask them to determine if the sketch of the inverse is also a function.
8. Repeat steps 3-7 for the graph of y=x^2. Make sure that students realize that not all functions have inverse functions (the graphs of the inverse also pass the vertical line test).
9. Allow students to work in pairs or groups to complete the Is the Inverse a Function? Worksheet. Students will sort functions into two categories: those that have inverse functions and those that don't. They will make a list of characteristics for the functions with inverse functions as well as for those that do not have inverse functions. You may need to lead students to the discovery that the repeated y-values on the inverse of non-functions were repeated x-values on the original function.
10. Allow students to present their findings. If students do not discover the horizontal line test for determining if a function is a one-to-one function (it has an inverse function) share this with them.
11. Now ask student pairs/groups to apply their findings to a table of values. Ask them to explain how to find the points on the inverse and how would they know if the function was a one-to-one function.
12. Circulate among the groups to address problems and misconceptions.
13. Allow students to present their findings.
Time allotment: 5-10 minutes
Give students the Quick Check exit slip. They will find the inverse of the given functions and decide if the function is one-to-one.