Lesson Plan

Collaboratively Exploring Reciprocal Functions and Transformations

Students develop an understanding for how the equations relates to it graph.
Philip K.
Classroom teacher
The Academy of Science and Entrepreneurship
Bloomington, IN
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My Grades 9, 10, 11, 12
My Subjects Math
EdTech Mentor

In this lesson flow, students begin by using a table to generate points and sketch the graph for the equation y = 1/x.  From here, students use online graphing technology to explore how the values of a, h and k transform the graph for y = a/(x-h) + k.  Students explore individually, make conjectures and then use online discussion tools to share and discuss their findings.  Once a level of consensus is achieved, students apply and deepen their understanding by matching various graphs with the equations that create them.  Finally, students take a brief "exit ticket" quiz to assess their level of understanding.

Students will be able to...

  • Identify Transformations to a parent function using values from the equation.
  • Determine the horizontal and vertical asymptotes of the graph.
  • Match a graph with the equation that would create it.
  • Construct viable arguments and critique the reasoning of others.
Grades 10 - 12
All Notes
Teacher Notes
Student Notes

1 Introduction

Activity: Drawing

Have students use an x,y table of values to sketch the graph of the equation y = 1/x using x values of {-10, -5, -4, -3, -2,-1, -1/2, 0, 1/2, 1, 2, 3, 4, 5, 10}

Highlight and discuss the following:

  • What happens at x = 0?
  • What will happen as x approaches 0? (Use the fractional values of x as a beginning for the conversation)
  • What happens as |x| grows large?
  • Vertical and Horizontal Asymptotes

Students should come away from this with a basic understanding of the shape of the graph (2 branches) and its asymptotes.

2 Exploring Transformations

Individually (or in pairs), have students access the following desmos graph.


The equation is in the form y = a/(x-h) + k, with sliders to change the values of a, h and k.  

Students should adjust the values one at a time, noting the corresponding changes in the graph.

Students should record their observations and conjectures to be shared in the next step.

3 Discussion of Findings

Using one of the suggested online discussion tools (preferrably Collaborize which better supports various threads simultaneously and allows students to more easily respond to and vote for entries from their peers), have students respond to the following prompt:

How did the values in the equation affect the graph?

  • a - value
  • h - value
  • k - value

Encourage students to react and respond to one anothers conmments and conclusions.

Attempt to use student findings to develop consensus on how the values in the equation allow one to identify key aspects of the graph.  "Boil Down" student findings into clear and concise conclusions about the specific effect(s) of a, h and k values.

4 Independent Practice

Free, Paid

Have students spend some time reviewing the following Quizlet Cards which pair a given equation with its graph.

Students can review them in a flashcard style format, or in a more game-like setting using the matching game or the space race feature.

Remind students that the goal is to deepen their understanding of how the number values in the equation relate to the key features of the graph.

5 Exit Ticket

As a closing move, students should take the following Socrative Quiz (SOC #: 16138506) to measure their level of understanding for the goals of the lesson.

(To access quiz, log in to socrative teacher and go to "Manage Quizzes" then select "Import Quiz" and enter the number from above.)