Unit 6, Lesson 3
Equations for Functions
Accelerated 7
3.1
Warm-up
Fill in the table of input-output pairs for the given rule. Write an algebraic expression for the rule in the box in the diagram.
| input | output |
|---|---|
| 8 | |
| 2.2 | |
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Fill in the table of input-output pairs for the given rule. Write an algebraic expression for the rule in the box in the diagram.
| input | output |
|---|---|
| 8 | |
| 2.2 | |
Record your answers to these questions in the table provided.
| description | a | b | c | d |
|---|---|---|---|---|
| diagram | ||||
| equation | ||||
| input = 5 output = ? |
||||
| independent variable |
||||
| dependent variable |
We can sometimes represent functions with equations. For example, the area, , of a circle is a function of the radius, , and we can express this with this equation:
We can also draw a diagram to represent this function:
In this case, we think of the radius, , as the input and the area of the circle, , as the output. For example, if the input is a radius of 10 cm, then the output is an area of cm2, or about 314 cm2. Because this is a function, we can find the area, , for any given radius, .
Since is the input, we say that it is the independent variable, and since is the output, we say that it is the dependent variable.
We sometimes get to choose which variable is the independent variable in the equation. For example, if we know that
then we can think of as a function of and write
or we can think of as a function of and write
A dependent variable represents the output of a function.
For example, someone needs to buy 20 pieces of fruit and decides to buy apples and bananas. If they select the number of apples first, the equation shows the number of bananas they 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 someone needs to buy 20 pieces of fruit and decides to buy some apples and bananas. If they select the number of apples first, the equation shows the number of bananas they can buy. The number of apples is the independent variable because any number can be chosen for it.