## Functions: Overview

Functions are rules that describe the relationship between two variables. Common examples of functional relationships include the relationship between the distance traveled at a certain rate over a period of time, the relationship between the unit cost of an item and the cost for multiple numbers of that item, and the relationship between temperatures in degrees Celsius and degrees Fahrenheit.

Functions are commonly expressed as equations. When you know the value of one variable, you can use a rule or an equation to find the value of the other variable. For example, in the function *y* = 2*x* + 6, if you know that *x* = 3, you can calculate that the value of *y* is 12. It is important for students to understand that for each value of a variable, there is exactly one related value for the other variable. In the previous example when *x* = 3, *y* can **only** equal 12.

The two variables in a function have names that explain how they relate to one another. One variable, usually *x,* is known as the input or independent variable. This is the variable upon which the operation or operations are performed. The other variable, usually *y,* is called the output or dependent variable. The value of this variable is dependent upon the value of the independent variable.

Functions are also displayed or described in the form of function tables or graphs. A function table is a table of ordered pairs following the rule for that function. Function tables can be made up of an infinite number of ordered pairs. A function table for the equation *y* = 2*x* + 6 is shown below.

x |
y |

1 | 8 |

2 | 10 |

3 | 12 |

4 | 14 |

Once you have created a function table, you can graph the ordered pairs from the table on the coordinate plane. When graphing the values in a function table, the value of the input variable is the *x*-coordinate and the corresponding value of the output variable is the *y*-coordinate. The graph for the function *y* = 2*x* + 6 is shown here.

By extending the line on the graph, students can find other values for the variables in a function.

Students were introduced to the concept of functional relationships in Grade 1 through the use of simple input-output tables. Grade 4 students worked with functions involving variables and two-step functions (functions in which more than one operation is performed). They also learned how to graph functions.

Earlier in Grade 5, students worked with functions relating to patterns and involving a single operation, either addition or subtraction. A pattern is a design or an arrangement of numbers that models a general rule. Each design or number in the pattern is called a *term*. In the pattern below, the first term shows 1 circle and 0 squares; the second term shows 2 circles and 1 square; the third term shows 3 circles and 2 squares.

By identifying the pattern, students can determine how many circles or squares are needed for any term. In this pattern the number of squares is always one less than the number of the term. This relationship can be written as an equation. So if *x* stands for the term, and *y* stands for the number of squares, the function for the squares in this pattern is *y* = *x* − 1. There is a different function and therefore a different equation to describe the relationship between the term and the number of circles. Students should understand that in multishape or multicolor patterns, each shape or color may have its own function.

In the current topic, students are working with one-step and two-step functions involving addition, subtraction, multiplication, and division. Later, students will work with functions involving integers and they will learn to graph these functions on the coordinate plane.