So far we have used input-output diagrams and descriptions of the rules to describe functions. This is the first of five lessons that introduces and connects the different ways in which we represent functions in mathematics: verbal descriptions, equations, tables, and graphs. In this lesson students transition from input-output diagrams and descriptions of rules to equations.
This lesson also introduces the use of independent and dependent variables in the context of functions. For an equation that relates two quantities, it is sometimes possible to write either of the variables as a function of the other. For example, in the activity Dimes and Quarters, we can choose either the number of quarters or the number of dimes to be the independent variable. If we know the number of quarters and have questions about the number of dimes, then this would be a reason to choose the number of quarters as the independent variable.
- Calculate the output of a function for a given input using an equation in two variables, and interpret (orally and in writing) the output in context.
- Create an equation that represents a function rule.
- Determine (orally and in writing) the independent and dependent variables of a function, and explain (orally) the reasoning.
Let’s find outputs from equations.
- I can find the output of a function when I know the input.
- I can name the independent and dependent variables for a given function and represent the function with an equation.
A dependent variable represents the output of a function.
For example, suppose we need to buy 20 pieces of fruit and decide to buy apples and bananas. If we select the number of apples first, the equation \(b=20-a\) shows the number of bananas we can buy. The number of bananas is the dependent variable because it depends on the number of apples.
An independent variable represents the input of a function.
For example, suppose we need to buy 20 pieces of fruit and decide to buy some apples and bananas. If we select the number of apples first, the equation \(b=20-a\) shows the number of bananas we can buy. The number of apples is the independent variable because we can choose any number for it.
A radius is a line segment that goes from the center to the edge of a circle. A radius can go in any direction. Every radius of the circle is the same length. We also use the word radius to mean the length of this segment.
For example, \(r\) is the radius of this circle with center \(O\).