Analyzing a Graph
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:
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
Explain why each function can or cannot be used to model the given data set.
a. 𝑓(𝑥) = 3𝑥 + 5
b. 𝑓(𝑥) = −(𝑥 − 2)2 + 9
c. 𝑓(𝑥) = −𝑥2 + 4𝑥 – 5
d. 𝑓(𝑥) = 3x + 4
e. 𝑓(𝑥) = (𝑥 −2)2 + 9
f. 𝑓(𝑥) = −(𝑥 + 1)(𝑥 − 5)
Match each table below to the function and the context, and explain how you made your decision.
𝑓(𝑥) = 12𝑥
ℎ(𝑥) = −9|𝑥 − 3| + 27
𝑔(𝑥) = −(𝑥)(𝑥 − 6)
𝑝(𝑥) = 2x
𝑞(𝑥) = −16𝑥
2 + 30𝑥 + 160
The following methods can be used to determine the appropriate model for a given data set as a linear, quadratic, or exponential function:
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