Recognizing Linear Functions

The rate of change is a ratio that describes, on average, how much one quantity changes with respect to a change in another quantity. Let's find how much the values in each column changes when we go from one data point to the next.

To find the rate of change we will use the following formula. $\text{rate of change }=\dfrac{\text{change in }y}{\text{change in }x}$ Let's find the rate of change between the first two rows. Here $y$ is represented by the height of the bamboo and $x$ by which day it is. The rate of change between the first two rows is $0.8.$ We can follow the same process to find the rate of change between the second and the third row in the table.

Changes between rows	$\dfrac{\text{change in } y}{\text{change in }x}$	Rate of change
$1^{\text{st}}\rightarrow2^{\text{nd}}$	$\dfrac{0.8}{1}$	$0.8$
$2^{\text{nd}}\rightarrow3^{\text{rd}}$	$\dfrac{0.8}{1}$	$0.8$

As we can see, the rate of change is the same for both pairs of points. Therefore, the function represented by the given table is linear. Since Don measures the height of the bamboo at the same time every day we can conclude that it is growing at a rate of $0.8$ meters per day.