Interpreting Rates of Change

A rate of change is the ratio between the change in one quantity to the change in another quantity. Linear relationships have a constant rate of change.

The tile pattern below is growing by three tiles per figure. Therefore, the tile pattern has a growth rate of 3.

A runner travels $\SI{80}{feet}$ in $\SI{8}{seconds}$. What is the runner's rate of change?

Answer

The runner is traveling $\dfrac{\SI{80}{feet}}{\SI{8}{seconds}}$ or $\SI{10}{feet.per.second}$.

Is the relationship between the inputs and outputs below linear?

Input	Output
0	11
3	5
5	1
8	-5

Answer

If the rate of change between pairs of inputs and outputs in constant, then the relationship is linear.

The rate of change between the pairs $(0,11)$ and $(3,5)$ is $\dfrac{11-5}{0-} = \dfrac{6}{-3}=-2.$

The rate of change between the pairs $(3,5)$ and $(5,1)$ is $\dfrac{5-1}{3-5} = \dfrac{4}{-2}=-2.$

The rate of change between the pairs $(5,1)$ and $(8,-5)$ is $\dfrac{1-(-5)}{5-8} = \dfrac{6}{-3}=-2.$

The relationship is linear with a rate of change of -2.

The slope of a line shows the rate of change in a linear relationship.

For example, the graph below shows a rate of change of 10 liters per second. The slope of the line is $\frac{10}{1}=10.$

Which car has the greatest gas mileage?

Answer

Line A is the steepest so it has the greatest rate of change so car A has the greatest gas mileage.

We can see that car A travels about 50 miles on one gallon of gas, car B travels about 30 miles on one gallon of gas, and car C travels about 10 miles on one gallon of gas.

Equations of lines in the form $y=mx+b$ represent linear functions with constant rates of change. The rate of change in the relationship is represented by $m.$

The equation $y=5,000x+12,0000$ represents the total number of miles on Zen's car, $y,$ each year that she owned it, $x.$ How many miles does Zen drive per year?

Answer

The growth rate in this equation is 5,000. Therefore, Zen drives 5000 miles per year. Zen's car had 12,000 miles on it when she purchased it.

In pattern A, the number of dots, $y,$ in figure $x$ is given by $y=6x+7.$

In pattern B the number of dots, $y,$ in figure $x$ is given by $y=7x+6.$

Which dot pattern is growing more quickly?

Answer

In pattern A, the growth rate is 6. In pattern B, the growth rate is 7. Therefore, pattern B is growing more quickly.