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Here are a few recommended readings before getting started with this lesson.
Here are a few practice exercises before getting started with this lesson.
Dominika is curious about the amount of time she spends going to and from school. One day, she decides to measure the time it takes her to get to school on foot and compare it to the amount of time it takes her to arrive home by bus. She drew the following graph for her trip to school.
In order to understand the relationship between two quantities and make predictions about their behavior, it is important to be able to compare and quantify how one changes with respect to the other. These ideas define the concept of rate of change.
The rate of change ROC is a ratio used to compare how a variable changes in relation to another variable. It is determined by dividing the change in the output variable $y$ by the change in the input variable $x.$ For any ordered pairs $(x_{1},y_{1})$ and $(x_{2},y_{2}),$ the rate of change is calculated using the following formula.
The Greek letter $Δ$ (Delta) is commonly used to represent a difference. This leads to an alternative way of writing the formula for ROC.
$Rate of Change=ΔxΔy $
The units of a rate of change are the ratio of output units to the input units, meaning that they are derived units. Interpreting the rate of change depends on the context.
Output Units  Input Units  Rate of Change Units  Possible Interpretation 

meters  seconds  $secondmeters $  Car speed over a certain period of time 
bacteria  hours  $hourbacteria $  Growth rate of bacteria in an experiment 
U.S. dollars  hours  $hourU.S.dollars $  Worker's hourly wage 
As mentioned above, the rate of change is a quantity that compares the change in the output variable to the input variable. These are also known as the dependent and independent variables, respectively.
Diego is taking the bus back to his hometown. During the trip, he sees that there are signs by the side of the road that indicate the distance from the city.
He keeps track of some of the signs and notes the time taken to reach them. He puts the information together using a table of values.
Time (minutes)  Distance (kilometers) 

$0$  $0$ 
$10$  $15$ 
$20$  $30$ 
$30$  $45$ 
$40$  $60$ 
$50$  $75$ 
$60$  $90$ 
Substitute $(10,15)$ & $(20,30)$
Rate of Change $=t_{2}−t_{1}d_{2}−d_{1} $  

Times  Substitute  Evaluate 
$1020$ min  $20min−10min30km−15km $  $1.5minkm $ 
$060$ min  $60min−0min90km−0km $  $1.5minkm $ 
Substitute $(0,0)$ & $(2,50)$
Rate of Change $=x_{2}−x_{1}y_{2}−y_{1} $  

Points  Substitute  Evaluate 
$(0,0),(2,50)$  $2−050−0 $  $25$ 
$(2,50),(4,100)$  $4−2100−50 $  $25$ 
Linear Equation: $y=2x+3$  

$x$  Substitute  Point 
$x=1$  $yy =50(1)+30=80 $

$(1,80)$ 
$x=3$  $yy =50(3)+30=180 $

$(3,180)$ 
$x=4$  $yy =50(4)+30=230 $

$(4,230)$ 
$x=7$  $yy =50(7)+30=380 $

$(7,380)$ 
Now that the points are known, all that is left to do is to repeat the same procedure as in Part B. The following table summarizes the results for the pair of points of interest.
Rate of Change $=x_{2}−x_{1}y_{2}−y_{1} $  

Points  Substitute  Evaluate 
$(1,80),(3,180)$  $3−1180−80 $  $50$ 
$(4,230),(7,380)$  $7−4380−230 $  $50$ 
$(1,80),(7,380)$  $7−1380−80 $  $50$ 
It is interesting to note that the rate of change is the same between all the pairs of points used. This occurred in Parts A and B as well.
The following applet shows different representations for the data of two variables. Practice finding the rate of change between the indicated values. If the answer is not an integer number, round it to two decimal places.
$y_{1}=mx_{1}+b$, $y_{2}=mx_{2}+b$
Distribute $1$
Subtract terms
Factor out $m$
$ca⋅b =a⋅cb $
$aa =1$
$a⋅1=a$
Consider a set of data points for which the rate of change between every pair of consecutive points is a constant value $m.$ Consider two consecutive points $(x_{1},y_{1})$ and $(x_{2},y_{2}).$
Rate of Change  Rewrite 

$x_{2}−x_{1}y_{2}−y_{1} =m$  $y_{2}−y_{1}=m(x_{2}−x_{1})$ 
Note that since the rate of change between every pair of consecutive points is constant, they all can be rewritten in a similar way.
Rate of Change  Rewrite 

$x_{2}−x_{1}y_{2}−y_{1} =m$  $y_{2}−y_{1}=m(x_{2}−x_{1})$ 
$x_{3}−x_{2}y_{3}−y_{2} =m$  $y_{3}−y_{2}=m(x_{3}−x_{2})$ 
$⋮$  $⋮$ 
$x_{n}−x_{n−1}y_{n}−y_{n−1} =m$  $y_{n}−y_{n−1}=m(x_{n}−x_{n−1})$ 
$y_{3}−y_{2}=m(x_{3}−x_{2})$, $y_{2}−y_{1}=m(x_{2}−x_{1})$
$y_{4}−$