Chapter 8, Problem 8PT

To determine

To find: the rate of change and determine whether a proportional linear relationship exists or not.

Answer to Problem 8PT

The linear relationship is proportional

The rate of change for the linear function is 2 steps stones per meter

Explanation of Solution

Given:

  

Calculation:

Consider three points from the graph that shows relationship between number of stepping stones and length of path: $\left(2,4\right),\left(3,6\right)$ and $\left(5,10\right)$

In order to find constant rate of change, take any two points and use the formula for rate of change.

  $\text{Rate of change}=\frac{\text{change}\text{in}y}{\text{change}\text{in}x}$

It means rate of change is change in number of stepping stones with respect to change in length of path.

So, consider two points as: $\left(2,4\right)$ and $\left(5,10\right)$

Thus, constant rate of change is:

  $\begin{array}{c}\text{Rate of change}=\frac{10-4}{5-2}\\ =\frac{6}{3}\\ =2\end{array}$

Hence, rate of change for the linear function is 2 steps stones per meter

Linear relationships are called proportional if the ration of each non zero y -value compared to corresponding x - value is the same.

Now, it means that calculate ratio

  $\frac{\text{Number of step stones}}{\text{length}\text{of}\text{path}}$ for each paired values.

For the point $\left(2,4\right)$ the ratio is: $\frac{4}{2}=2$

And. for the point $\left(3,6\right)$ the ratio is: $\frac{6}{3}=2$

And also, for the point $\left(5,10\right)$ the ratio is: $\frac{10}{5}=2$

Hence, ratio $\frac{\text{number of step stones}}{\text{length of path}}$ is same foreach paired value.

Conclusion:

Thus, the linear relationship is proportional and the rate of change for the linear function is 2 steps stones per meter