## Learning Math: Patterns, Functions, and Algebra

# More Nonlinear Functions

## Investigate more non-linear functions, focusing on cyclic and reciprocal functions. Become familiar with inverse proportions and cyclic functions, develop an understanding of cyclic functions as repeating outputs, work with graphs, and explore contexts where inverse proportions and cyclic functions arise. Explore situations in which more than one function may fit a particular set of data.

### In This Session

**Part A:** Cyclic Functions

**Part B:** Inverse Proportions

**Part C:** Different Functions

**Homework**

In Session 7, we explored exponential and quadratic functions in tables, graphs, and real-life situations. We learned that exponential functions have constant ratios between successive outputs and that quadratic functions have constant second differences. This session continues the exploration of nonlinear functions, focusing on cyclic and inverse functions. **Note 1**

### Learning Objectives

This session introduces cyclic functions and inverse proportions. Through the activities in this session, we will:

- Become familiar with inverse proportions and cyclic functions
- Develop an understanding of cyclic functions as repeating outputs
- Work with graphs of inverse proportions and cyclic functions
- Explore contexts where inverse proportions and cyclic functions arise
- Explore situations in which more than one function may fit a particular set of data

### Key Terms

**Previously Introduced:**

**Direct Variation: **A direct variation is a relationship between inputs and outputs in which the ratio of inputs and outputs is always the same.

**Linear Relationship: **A constant rate produces a linear relationship between two variables.

**Exponential Function: **In an exponential function the independent variable is an exponent in an equation. Functions like y = 2^{x} and y = 10(.5)^{x} are exponential functions. An exponential function has a constant ratio between successive outputs. For example, in y = 2^{x}, each time x grows by 1, y is multiplied by 2.

**Quadratic Function:** A quadratic function is a function in which the independent variable is squared. The function y = x^{2} is the most basic quadratic function. All quadratic functions fit the form y = Ax^{2} + Bx + C, where A, B, and C can be any real number (although A cannot be zero). The graph of a quadratic function is called a parabola.

**New in This Session:**

**Cyclic Function: **A cyclic function is a function whose outputs repeat in a cycle. A traffic light is an example of a cyclic function. Cyclic functions have important applications in astronomy and the study of wave motion, including sound and light waves.

**Amplitude: **The amplitude of a cyclic function measures the height of the cyclic function, relative to its average. The amplitude can be determined by the formula: amplitude = (high – low) / 2.

**Period: **The period of a cyclic function measures how long it takes to complete one cycle of the function. For example, the period of time it takes for Earth to revolve around the sun is one year.

**Inverse Proportion: **An inverse proportion is a function in which the output changes in a reciprocal relationship to the input. In other words, if the input doubles, the output is cut in half.

**Inverse Variation: **An inverse variation is a relationship between two variables in which a change in one variable results in an inverse, or reciprocal, change in the other. If one variable is multiplied by 3, the other is divided by 3. Forms of inverse variation are A x B = k or B = k / A, where k is a constant number and A and B are the two variables.

**Modular Arithmetic:** In modular arithmetic, the result of a calculation is the remainder when dividing by the modulus. For example, 33 mod 7 would give the remainder when 33 is divided by 7 — 33 divides evenly into 7 four times, leaving a remainder of 5. Therefore, 33 mod 7 = 5. An alternate form is to write 33 = 5 (mod 7). Modular arithmetic is sometimes referred to as remainder arithmetic, since its calculations are done with remainders.

### Notes

**Note 1**

This session builds on the exploration of nonlinear functions that we began in Session 7. The activities in this session will help us develop a better understanding of cyclic functions and inverse variation.

The most famous cyclic functions are the trigonometric functions — in particular, sine and cosine. Trigonometry is beyond the scope of this course, but we can still look at cycles as a kind of function, without worrying about closed or recursive formulas to describe them. The general method of finding formulas is to start with a sine curve and to find a way to make the curve fit the data. This is similar to finding a “line of best fit” by changing the slope and intercept on the linear equation y = x.

**Materials Needed:** graph paper

**Review**

**Groups:** Discuss questions about the homework. In particular, talk about Problem H2 and share the methods for solving it. Note that there is an algebraic equivalent to “undoing an exponent”: taking a logarithm. (Logarithms are beyond the scope of this course, but it’s important to acknowledge that such an inverse operation exists.)

**Groups:** Review the list of functions from the previous session. At least two more kinds of functions will be added to that list during this session.