Modeling a Context from Data

New York State Common Core Math Algebra I, Module 5, Lesson 6

Worksheets for Algebra 1

Student Outcomes

• Students write equations to model data from tables, which can be represented with linear, quadratic, or exponential functions, including several from Lessons 4 and 5. They recognize when a set of data can be modeled with a linear, exponential, or quadratic function and create the equation that models the data.
• Students interpret the function in terms of the context in which it is presented, make predictions based on the model, and use an appropriate level of precision for reporting results and solutions.

Modeling a Context from Data

Classwork

Opening Exercise

a. Identify the type of function that each table appears to represent (e.g., quadratic, linear, exponential, square root).
b. Explain how you were able to identify the function.
c. Find the symbolic representation of the function.
d. Plot the graphs of your data

Example 1

Enrique is a biologist who has been monitoring the population of a rare fish in Lake Placid. He has tracked the population for 5 years and has come up with the following estimates:
Create a graph and a function to model this situation, and use it to predict (assuming the trend continues) when the fish population will be gone from the Lake Placid ecosystem. Verify your results, and explain the limitations of each model.

• Show Video for Whole Lesson

Exercises

1. Bella is a BMX bike racer and wants to identify the relationship between her bike’s weight and the height of jumps (a category she gets judged on when racing). On a practice course, she tests out 7 bike models with different weights and comes up with the following data.
   a. Bella is sponsored by Twilight Bikes and must ride a 32 lb bike. What can she expect her jump height to be?
   b. Bella asks the bike engineers at Twilight to make the lightest bike possible. They tell her the lightest functional bike they could make is 10 lb. Based on this data, what is the highest she should expect to jump if she only uses Twilight bikes?
   c. What is the maximum weight of a bike if Bella’s jumps have to be at least 2 ft high during a race?
2. The concentration of medicine in a patient’s blood as time passes is recorded in the table below
   a. The patient cannot be active while the medicine is in his blood. How long, to the nearest minute, must the patient remain inactive? What are the limitations of your model(s)?
   b. What is the highest concentration of medicine in the patient’s blood?
3. A student is conducting an experiment, and as time passes, the number of cells in the experiment decreases. How many cells will there be after 16 minutes?

Lesson Summary

When given a data set, strategies that could be used to determine the type of function that describes the relationship between the data are

• Determine the variables involved and plot the points.
• After making sure the 𝑥-values are given at regular intervals, look for common differences between the data points—first and second.
• Determine the type of sequence the data models first, and then use the general form of the function equation to find the parameters for the symbolic representation of the function.

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