# Lesson 4

### Lesson Narrative

In an earlier unit, students compared linear and exponential growth and observed that exponential growth eventually overtakes linear growth (even if a quantity showing linear growth starts out with a much greater value). They examined this phenomenon and then observed that it will always happen at a large enough input value.

In this lesson, students investigate how quantities that grow quadratically compare to those that grow exponentially. They discover and reason that increasing exponential functions also eventually surpass increasing quadratic functions. By examining successive quotients for each type of function, students see that the outputs of quadratic functions are not multiplied by the same factor each time the input increases by one. In fact, these successive quotients get smaller as the inputs increase, while the outputs of the exponential function have the same multiplier. As they compare the two types of functions, they develop their understanding of quadratic expressions and how shape of the graph differs between the two types of functions.

### Learning Goals

Teacher Facing

• Use graphs, tables, and calculations to show that exponential functions eventually overtake quadratic functions.

### Student Facing

• Let’s compare quadratic and exponential changes and see which one grows faster.

### 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.)

### Student Facing

• I can explain using graphs, tables, or calculations that exponential functions eventually grow faster than quadratic functions.

Building On