# Lesson 4

Representing Functions at Rational Inputs

### Lesson Narrative

Building on the previous lesson, students continue to reason about growth and decay factors over fractional intervals and to write functions in terms of these intervals. A focus of this lesson is on reasoning about the growth or decay factor for different time intervals and what the factors mean in context.

For example, the function \(f(d) = 5,\!000 \boldcdot (1.5)^d\) might represent a population in a city \(d\) decades since the population was 5,000. While the population grows by 50% each decade, the annual growth factor would be \((1.5)^\frac{1}{10}\), or a bit more than 4% per year, because each year is \(\frac{1}{10}\) of a decade. If \(t\) is time in years, then we can also say that the function \(g(t) = 5,\!000 \boldcdot \left(\sqrt[10]{1.5}\right)^t\) represents the population of the city.

Students construct viable arguments to articulate why finding the growth or decay rate for a fractional interval means extracting the root of the growth or decay rate for the whole interval instead of dividing the rate (MP3). Next, they use properties of exponents to interpret and transform expressions that represent decay. The work requires them to carefully consider the meaning of the decay factor in the given context (MP2) and to attend to the units of the input variable (MP6).

### Learning Goals

Teacher Facing

- Interpret fractional inputs for exponential functions in context.
- Understand that the growth (or decay) factor for an input interval of $\frac{1}{n}$ is $b^{\frac{1}{n}}$ and not $\frac{b}{n}$ when $b$ is the growth (or decay) factor for an interval of length 1 and $n$ is the number of intervals.

### Student Facing

- Let’s find how quantities are growing or decaying over fractional intervals of time.

### Learning Targets

### Student Facing

- I understand how to calculate a growth or decay factor of an exponential function for different input intervals.