# Lesson 19

Which One Changes Faster?

### Lesson Narrative

This lesson returns to a theme from the beginning of the unit, revisiting the fact that exponential functions grow more quickly, eventually, than linear functions. The first activity compares simple interest (linear growth) with compound interest (exponential growth). Students examine tables and graphs and see that the exponential function quickly overtakes the linear function. In the second activity, the exponential function is deliberately chosen to grow slowly over a large domain, making it less clear whether or not the exponential function will overtake the linear function.

The second activity provides an opportunity for students to strategically use technology (MP5), whether they make a graph (for which they will need to think carefully about the domain and range) or continue to tabulate explicit values of the two functions (likely with the aid of a calculator for the exponential function). This lesson provides multiple opportunities for students to justify their reasoning and critique the reasoning of others (MP3).

### Learning Goals

Teacher Facing

• Use graphs and calculations to show that a quantity that increases exponentially will eventually surpass one that increases linearly.
• Use tables, calculations, and graphs to compare growth rates of linear and exponential functions.

### Student Facing

Let's compare linear and exponential functions as they continue to increase.

### Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

Be prepared to display a graph for all to see using the embedded Desmos applet or other graphing technology.

### Student Facing

• I can use tables, calculations, and graphs to compare growth rates of linear and exponential functions and predict how the quantities change eventually.