Algebra 2 Question of the Day

1. 6% growth per year (correct answer)
2. 1.06% growth per year
3. 6% decay per year
4. It is linear growth because the exponent is $t$

Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! In the model A(t) = 1200(1.06)^t, the base is 1.06. Since 1.06 > 1, this represents exponential growth. The growth rate is 1.06 - 1 = 0.06 = 6% per year. Choice A correctly identifies 6% growth per year. Choice B incorrectly interprets 1.06 as 1.06% growth (it's actually the growth factor, not the percent), while choice C wrongly claims decay when the base is greater than 1, and choice D misunderstands the exponential nature of the function. The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms!