Exploring Exponential Functions: Graphing, Transformations, and Growth/Decay

Introduction – Warm-Up: Productive Struggle Question (5 minutes)

Total Time: 5 minutes

Pose the question: 'If y = 2^x represents an exponential function, how can we represent a function that grows faster as x increases? Graph your initial thoughts.' Allow students a moment to think independently, then discuss in pairs to share their ideas. Encourage them to consider transformations such as vertical and horizontal shifts.

Core Activity – Transformations and Graphing (10 minutes)

Total Time: 10 minutes

Provide students with different functions: e.g., y = 2^(x-1) + 3 (horizontal shift and vertical shift), y = 0.5^(x) (decay function), and y = 3(2^x) (vertical stretch).

In groups, have them graph these functions on graph paper or using graphing software. Each group must identify key transformations and characteristics (growth vs. decay) for each function. Allow students to showcase their graphs to the class and discuss their observations, highlighting how changes in the equation affect the graph.

Assessment – Exit Ticket: Short Constructed Response (5 minutes)

Total Time: 5 minutes

As an exit ticket, ask students to respond to the following prompt: 'Choose one of the functions we explored today. Describe the transformations you identified and explain how they affect the function’s growth or decay. Provide an example of real-world application for a similar exponential function.' This short constructed response will assess students' understanding of transformations and their ability to articulate mathematical concepts.