# Input-Output Tables

In order to understand the basics of graphing functions, you might start to encounter input-output tables. An input-output table is a visual aid that can help you find ordered pairs for a function when the underlying rule (or function) is known. Here is an example:

x | 0 | 1 | 2 | 3 | 4 |

$y=3x+1$ | 1 | 4 | 7 | 10 | 13 |

The function rule for this input-output table is $y=3x+1$ , meaning that we can plug any real value x into that equation to find its corresponding y value. Then, we have (x,y) coordinates that the graph of the function must pass through.

Generally speaking, you'll want to start plugging in x values starting with the simplest to calculate. For instance, 0 is the first input above because it's easy to calculate $3\left(0\right)+1=1$ for the first output. We then go up by a whole number for each subsequent input even though we could incorporate decimals or fractions if we wanted to.

Once we have all five ordered pairs, we can graph the function as shown below:

Remember to extend your line in both directions since you will never be able to plot every point on a function!

## What else can I do with input-output tables?

While input-output tables are usually used to help graph functions, you may also see one that gives you the x and y values and asks you to determine a corresponding function rule. Consider the following example:

x | 10 | 16 | 26 | 100 | 200 |

y | 4 | 7 | 12 | 49 | 99 |

The easiest way to approach problems like this is to pick an ordered pair and try to identify the relationship between them. Once you do, plug other values into the resulting function to see if they work too. For instance, $y=x-6$ works for the first pair since $4=10-6$ is accurate. However, that's not the function we're looking for since $7\ne 16-6$ . Instead, we need to look at what happens when we change the value of x and look at the corresponding value of y. When x went from 10 to 16 (a change of 6 units) y went from 4 to 7 (a change of 3 units) so it looks like from these two data points y scales at $\frac{1}{2}$ the value of x, but is also shifted by 6. So the formula might be $f\left(x\right)=\frac{1}{2}x-1$ . And sure enough, this also fits the rest of our data.

Importantly, you have to remember that any function determined this way is just a guess. Many different functions go through any given set of finite ordered pairs, and you won't be able to say for sure what the function is through guesswork alone. Still, there will typically be a fairly obvious answer whenever you see problems like this.

## Input-output tables practice questions

a. Construct an input-output table for the function rule $y=2x+1$ . Include at least three x values.

x | 0 | 1 | 2 |

$y=2x+1$ | 1 | 3 | 5 |

b. Create an input-output table for the function rule $y=5x-3$ .

x | 0 | 1 | 2 |

$y=5x-3$ | -3 | 2 | 7 |

c. Name one function that would satisfy the input-output table below:

x | 1 | 5 | 50 | 100 | 10,000 |

y | 4 | 16 | 151 | 301 | 30,001 |

$y=3x+1$

## Topics related to the Input-Output Tables

Writing Number Patterns in Function Notation

## Flashcards covering the Input-Output Tables

Common Core: 5th Grade Math Flashcards

## Practice tests covering the Input-Output Tables

MAP 5th Grade Math Practice Tests

Common Core: 5th Grade Math Diagnostic Tests

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