Linear, Quadratic, & Exponential Models
Interpret expressions for functions in terms of the situation they model.
5. Interpret the parameters in a linear or exponential function in terms of context
Clarification:Tasks are limited to constructing linear and exponential functions in simple context (not multi-step). / Understandings and notes about standard:
- I can identify maximums and minimums of exponential functions
- I can identify the rate of change of linear and exponential functions to identify specific locations on a graph and make inferences for data not shown
- I can interpret function notation
- I can identify the difference between a linear and exponential function (graphically and algebraically)
- I can produce a table for a linear and exponential function
- I can graph linear and exponential functions
Sample Tasks:
F-LE Taxi!
Alignments to Content Standards
•Alignment: A-REI.D.10
•Alignment: F-LE.B.5
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Lauren keeps records of the distances she travels in a taxi and what she pays:
Distance,
d
, in miles / Fare,
F
, in dollars
3 / 8.25
5 / 12.75
11 / 26.25
- If you graph the ordered pairs (d,F) from the table, they lie on a line. How can you tell this without graphing them?
- Show that the linear function in part (a) has equation F=2.25d+1.5 .
F-LE US Population 1982-1988
Alignments to Content Standards
•Alignment: F-LE.B.5
•Alignment: F-LE.A.1.b
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The below table provides some U.S. Population data from 1982 to 1988:
U.S. Population 1982 – 1988
Year / Population
(in thousands) / Change in Population
(in thousands)
1982 / 231,664 / ----
1983 / 233,792 / 233,792 - 231,664 = 2,128
1984 / 235,825 / 2,033
1985 / 237,924 / 2,099
1986 / 240,133 / 2,209
1987 / 242,289 / 2,156
1988 / 244,499 / 2,210
Notice: The change in population from 1982 to 1983 is 2,128,000, which is recorded in thousands in the first row of the 3rd column. The other changes are computed similarly. All population numbers in the table are recorded in thousands.
Source:
- If we were to model the relationship between the U.S. population and the year, would a linear function be appropriate? Explain why or why not.
- Mike decides to use a linear function to model the relationship. He chooses 2,139, the average of the values in the 3rd column, for the slope. What meaning does this value have in the context of this model?
- Use Mike's model to predict the U.S. population in 1992.
Alignments to Content Standards
•Alignment: F-LE.B.5
•Alignment: F-LE.A.1.c
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A fisherman illegally introduces some fish into a lake, and they quickly propagate. The growth of the population of this new species (within a period of a few years) is modeled by
P(x)=5bx, wherexis the time in weeks following the introduction andbis a positive unknown base.
- Exactly how many fish did the fisherman release into the lake?
- Find b if you know the lake contains 33 fish after eight weeks. Show step-by-step work.