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Algebra / Linear functions Difficulty: Medium

The figure presents a 2-column table, with two rows of data, titled “Population of Greenleaf, Idaho.” The heading of the first column is “Year.” The heading of the second column is “Population.” The 2 rows of data are as follows. Row 1. Year, 2000. Population, 862. Row 2. Year, 2010. Population, 846

The table above shows the population of Greenleaf, Idaho, for the years 2000 and 2010. If the relationship between population and year is linear, which of the following functions P models the population of Greenleaf t years after 2000?

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Explanation

Choice A is correct. It is given that the relationship between population and year is linear; therefore, the function that models the population t years after 2000 is of the form $P of t equals, m, t plus b$ , where m is the slope and b is the population when $t equals 0$ . In the year 2000, $t equals 0$ . Therefore, $b equals 862$ . The slope is given by $m equals, the fraction with numerator P of 10, minus P of 0, and denominator 10 minus 0, end fraction, which equals the fraction with numerator 846 minus 862, and denominator 10 minus 0, end fraction, which equals negative 16 over 10, which equals negative 1 point 6$ . Therefore, $P of t equals, negative 1 point 6, t, plus 862$ , which is equivalent to the equation in choice A.

Choice B is incorrect and may be the result of incorrectly calculating the slope as just the change in the value of P. Choice C is incorrect and may be the result of the same error as in choice B, in addition to incorrectly using t to represent the year, instead of the number of years after 2000. Choice D is incorrect and may be the result of incorrectly using t to represent the year instead of the number of years after 2000.