Analyzing a Data Set

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New York State Common Core Math Algebra I, Module 1, Lesson 2

Worksheets for Algebra I, Module 1, Lesson 2 (pdf)

Student Outcomes

  • Students recognize linear, quadratic, and exponential functions when presented as a data set or sequence, and formulate a model based on the data.

Analyzing a Graph


Opening Exercise

When tables are used to model functions, we typically have just a few sample values of the function and therefore have to do some detective work to figure out what the function might be. Look at these three tables:

Example 1

Noam and Athena had an argument about whether it would take longer to get from NYC to Boston and back by car or by train. To settle their differences, they made separate, nonstop round trips from NYC to Boston. On the trip, at the end of each hour, both recorded the number of miles they had traveled from their starting points in NYC. The tables below show their travel times, in hours, and the distances from their starting points, in miles. The first table shows Noam’s travel time and distance from the starting point, and the second represents Athena’s. Use both data sets to justify your answers to the questions below.

a. Who do you think is driving, and who is riding the train? Explain your answer in the context of the problem.
b. According to the data, how far apart are Boston and New York City? Explain mathematically.
c. How long did it take each of them to make the round trip?
d. According to their collected data, which method of travel was faster?
e. What was the average rate of change for Athena for the interval from 3 to 4 hours? How might you explain that in the context of the problem?
f. Noam believes a quadratic function can be used as a model for both data sets. Do you agree? Use and describe the key features of the functions represented by the data sets to support your answer


  1. Explain why each function can or cannot be used to model the given data set.
    a. 𝑓(π‘₯) = 3π‘₯ + 5
    b. 𝑓(π‘₯) = βˆ’(π‘₯ βˆ’ 2)2 + 9
    c. 𝑓(π‘₯) = βˆ’π‘₯2 + 4π‘₯ – 5
    d. 𝑓(π‘₯) = 3x + 4
    e. 𝑓(π‘₯) = (π‘₯ βˆ’2)2 + 9
    f. 𝑓(π‘₯) = βˆ’(π‘₯ + 1)(π‘₯ βˆ’ 5)

  2. Match each table below to the function and the context, and explain how you made your decision.

𝑓(π‘₯) = 12π‘₯
β„Ž(π‘₯) = βˆ’9|π‘₯ βˆ’ 3| + 27
𝑔(π‘₯) = βˆ’(π‘₯)(π‘₯ βˆ’ 6)
𝑝(π‘₯) = 2x
π‘ž(π‘₯) = βˆ’16π‘₯
2 + 30π‘₯ + 160


  1. The population of bacteria doubled every month, and the total population vs. time was recorded.
  2. A ball was launched upward from the top of a building, and the vertical distance of the ball from the ground vs. time was recorded.
  3. The height of a certain animal’s vertical leap was recorded at regular time intervals of one second; the animal returned to ground level after six seconds.
  4. Melvin saves the same amount of money every month. The total amount saved after each month was recorded.
  5. Chris ran at a constant rate on a straight-line path and then returned at the same rate. His distance from his starting point was recorded at regular time intervals.

Lesson Summary

The following methods can be used to determine the appropriate model for a given data set as a linear, quadratic, or exponential function:

  • If the first difference is constant, then the data set could be modeled by a linear function.
  • If the second difference is constant, then the data set could be modeled by a quadratic function.
  • If the subsequent 𝑦-values are multiplied by a constant, then the data set could be modeled by an exponential function.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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