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We want to know if the function described in the following table is a linear function.

$x$ | $y$ |
---|---|

$2$ | $4$ |

$3$ | $8$ |

$4$ | $12$ |

$5$ | $16$ |

We can check the rate of change algebraically by noting the change in $x$ and change in $y$ between each of the data points.

For linear functions the rate of change is constant between any points of the function. We can find the rate of change between each row by dividing the change in $y$ by the change in $x.$ $rate of change=change inxchange iny $ Let's find the rate of change between the first two rows. The rate of change between the first two rows is $4.$ We can follow the same process for the other given data points.Changes between rows | $change inxchange iny $ | Rate of change |
---|---|---|

$1_{st}→2_{nd}$ | $14 $ | $4$ |

$2_{nd}→3_{rd}$ | $14 $ | $4$ |

$3_{rd}→4_{th}$ | $14 $ | $4$ |

As we can see, the rate of change is constant. Therefore, we can conclude that the function is linear.

b

If the data points in the table represents a linear function the points will line up when plotted in a coordinate plane. Let's plot the data points as $(x,y)$ coordinate pairs.

If the function is linear, connecting these points will form a straight line. Otherwise, we will have shown that the function is nonlinear. Let's connect each of our points with a straight edge and observe the result.

It looks like they have not lined up. To be safe we will connect the first and the last point with a separate straight edge.

Since the last straight edge we drew does not coincide with the three other we can conclude that the function is nonlinear.