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Recognizing Linear Functions

Recognizing Linear Functions 1.7 - Solution

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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. rate of change =change in ychange in x \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 yy is represented by the height of the bamboo and xx by which day it is.
rate of change =change in ychange in x\text{rate of change }=\dfrac{\text{change in }y}{\text{change in }x}
rate of change =0.81\text{rate of change }=\dfrac{{\color{#009600}{0.8}}}{{\color{#0000FF}{1}}}
rate of change =0.8\text{rate of change }=0.8
The rate of change between the first two rows is 0.8.0.8. 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 change in ychange in x\dfrac{\text{change in } y}{\text{change in }x} Rate of change
1st2nd1^{\text{st}}\rightarrow2^{\text{nd}} 0.81\dfrac{0.8}{1} 0.80.8
2nd3rd2^{\text{nd}}\rightarrow3^{\text{rd}} 0.81\dfrac{0.8}{1} 0.80.8

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 0.80.8 meters per day.