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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 (x1,y1) and (x2,y2), 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 =t2−t1d2−d1 | ||
---|---|---|
Times | Substitute | Evaluate |
10-20 min | 20 min−10 min30 km−15 km | 1.5minkm |
0-60 min | 60 min−0 min90 km−0 km | 1.5minkm |
Substitute (0,0) & (2,50)
Rate of Change =x2−x1y2−y1 | ||
---|---|---|
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 =x2−x1y2−y1 | ||
---|---|---|
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.
y1=mx1+b, y2=mx2+b
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 (x1,y1) and (x2,y2).
Rate of Change | Rewrite |
---|---|
x2−x1y2−y1=m | y2−y1=m(x2−x1) |
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 |
---|---|
x2−x1y2−y1=m | y2−y1=m(x2−x1) |
x3−x2y3−y2=m | y3−y2=m(x3−x2) |
⋮ | ⋮ |
xn−xn−1yn−yn−1=m | yn−yn−1=m(xn−xn−1) |
y3−y2=m(x3−x2), y2−y1=m(x2−x1)
y4−y3=m(x4−x3), y3−y1=m(x3−x1)
Diego's brother is very interested in crypotcurrencies and is telling Diego about how their values changed in the first weeks of the last year. One of them is of special interest to him because its graph is a straight line. He tried to print some information to show Diego, but his printer broke before it could print the complete tables of values for the cryptocurrencies.
Diego's brother wants to identify the table in which the values have a linear relationship. For each table, the left column represents the week number and the right column represents the value of the cryptocurrency in dollars. Which of the three tables best describes a linear relation?Linear relations grow by equal differences over equal intervals.
Since the relation represented in Table A increases by equal differences in equal intervals, the data can be modeled with a linear relation. Now, the same procedure will be applied to other tables.
Since the differences between consecutive y-values are not constant for equal intervals, Table B and Table C do not satisfy the condition. As such, they do not represent linear relations.
The following table shows the relationship between two variables, x and y. Look at the data carefully and determine whether the relationship between x and y can be described by linear equation.