Nonlinear Functions

Previously Introduced:

Closed-Form Description: A closed-form description of a pattern tells how to get from any input to its output, without having to know any previous outputs. A rule such as “take the input, triple it, and add two” is a closed-form description of a pattern.

Recursive Description: A recursive description of a pattern tells you how to proceed from one step to the next. For example, a recursive description might be, “Add two to the value of the output each time the input goes up by one.” The Fibonacci sequence, where each output is the sum of the two numbers before it, is a recursive description of a pattern.

Function: A function is any relationship between inputs and outputs in which each input leads to exactly one output. It is possible for a function to have more than one input that yields the same output.

Independent Variable: In a function where the value of variable A depends on the value of variable B, variable A is referred to as the dependent variable and variable B is referred to as the independent variable. In a table with inputs and outputs, the input is the independent variable and the output is the dependent variable.

Dependent Variable: In a function where the value of variable A depends on the value of variable B, variable A is referred to as the dependent variable and variable B is referred to as the independent variable. In a table with inputs and outputs, the input is the independent variable and the output is the dependent variable.

New in This Session:

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.

Base: In the exponential equation 8 = 2³, 2 is the base and 3 is the exponent.

Exponent: In the exponential equation 8 = 2³, 2 is the base and 3 is the exponent.

Exponential Growth Function: An exponential growth function is an exponential function that increases. The function y = 3^x is an exponential growth function; as x increases, y increases. Population growth and investment growing with interest are examples of exponential growth.

Exponential Decay Function: An exponential decay function is an exponential function that decreases. The function y = (.5)x is an exponential decay function; as x increases, y decreases. The decay of nuclear waste is an example of exponential decay.

Quadratic Function: A quadratic function is a function in which the independent variable is squared. The function y = x² is the most basic quadratic function. All quadratic functions fit the form y = Ax² + 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.

Parabola: The graph of a quadratic function is called a parabola.

Constant Differences: A table of common differences is formed by finding the difference between successive outputs. A table of second common differences can be formed by finding the difference between the differences. Common difference can be used to determine if a table of values comes from a linear or quadratic function. See Session 7, Part D for more information.

Figurate Number: A figurate number is a number of dots which form a geometric shape. If you make a square with 5 dots on a side, there will be 25 dots; this makes the number 25 a square number. If you make a triangle with 4 dots on a side, there will be 10 dots; this makes 10 a triangular number. Figurate numbers can be formed from pentagons, hexagons, cubes, pyramids, and other geometric shapes.

Note 1

In this session, we’ll explore nonlinear functions and situations in which these functions arise. People tend to think of functions mostly in terms of linear functions, but exponential, quadratic, and other nonlinear functions are also common in the world, and they’re also important to understand.

Another important idea in this section is that families of functions can be described by certain characteristics. For example, exponential functions are characterized by a constant ratio between successive outputs. Emphasizing these common characteristics helps us to think about functions as objects of study in and of themselves, and not just as rules that transform inputs into outputs.

Materials Needed: Computers with spreadsheet program, graph paper, toothpicks.

Groups: If it’s easy to move back and forth to computers, you may want to use the spreadsheet program throughout the session. If not, you may use it just for Part A, and work with a calculator and graph paper after that time.

Review
Groups: Discuss any questions that came up on the homework. Share solutions to the mobile problems. Talk about the thinking used in solving the problems. Did you think “algebraically?” Did you use any symbols to solve the problems?

Move into today’s session by reviewing what you know or remember about linear functions. Some key points are:

• There is a constant difference between successive outputs of a linear function
• Linear equations look like y = ax + b, where a and b are numbers and x is the input
• The graphs of linear functions are lines
• The slope of a linear function measures how much the output changes for each change of 1 in the input; in other words, it measures rise over run
• Slope is constant everywhere on a line
• There is only one linear equation that fits any two given points
• Rate problems are related to linear functions
• Direct variation is a kind of linear function that takes the form y = ax

Groups: You may want to record your comments on an overhead, flip chart, or blackboard.

Linear functions are important in mathematics, but there are a host of other mathematical functions used to model both abstract and real-world phenomena. This session will introduce just a few of those functions.

Groups: Brainstorm any functions that are not linear. You can leave the list on a board or paper posted in the room and add to it throughout the session.