Recognize Percent Growth or Decay - Algebra 2

Two phone plans are described below. Plan 1: You pay a $25 base fee plus $5 per gigabyte each month. Plan 2: You start at $25 and the bill increases by 5% each month. Which plan shows constant percent change (exponential) and which shows constant additive change (linear)?
Answer: Plan 2 is exponential; Plan 1 is linear 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! Plan 1 adds $5 per gigabyte—this is constant additive change (linear): total = 25 + 5×(gigabytes). Plan 2 increases by 5% each month—this is constant percent change (exponential): each month's bill = 1.05 × previous month's bill. Choice C correctly identifies Plan 2 as exponential (5% monthly increase) and Plan 1 as linear ($5 per gigabyte addition). Choice A reverses these classifications—Plan 1's fixed addition per gigabyte is linear, not exponential! 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!
A car's value is recorded at the end of each year: Year: 0, 1, 2, 3 Value ($): 25,000; 22,500; 20,250; 18,225 Classify: exponential growth, exponential decay, or neither? If exponential, what is the percent rate per year?
Answer: Exponential decay at 10% per year 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! Ratio test: 22,500/25,000 = 0.9, 20,250/22,500 = 0.9, 18,225/20,250 = 0.9, constant at 0.9, so exponential decay with b=0.9, percent rate (1 - 0.9)*100% = 10% per year. Choice C correctly identifies exponential decay at 10% per year through constant ratios and base analysis. Something like choice D might see the constant differences (-2,500; -2,250; -2,025) and think linear, but they're not constant, confirming it's not linear—excellent verification! 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!
A device's battery charge is measured every 30 minutes. Interval $n$ : 0, 1, 2, 3 Charge $C(n)$ (%): 100, 85, 72.25, 61.4125 Identify the constant percent rate of change per 30-minute interval and classify as growth or decay.
Answer: 15% decay per interval 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! Let's calculate the ratios: 85/100 = 0.85, 72.25/85 = 0.85, 61.4125/72.25 = 0.85. All ratios equal 0.85, confirming exponential decay with base 0.85. Since 0.85 = 1 - 0.15, the decay rate is 15% per 30-minute interval (the battery retains 85% of its charge each interval). Choice A correctly identifies 15% decay per interval. Choice B incorrectly interprets 0.85 as 85% decay—remember, 0.85 is the retention factor, meaning 15% is lost, not 85%! 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!
A medication amount in the bloodstream is modeled by $M(t)=80(0.92)^t$ , where $t$ is in hours. What percent does the amount change each hour, and is it growth or decay?
Answer: 8% decay per hour 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! From the function M(t) = 80 $(0.92)^t$ , the base is 0.92, indicating a constant multiplicative factor less than 1 each hour. Choice A correctly identifies 8% decay per hour through the base analysis (0.92 = 1 - 0.08). A distractor like choice B misinterprets the base as the percent decay directly, but it's the retention factor—remember, percent decay is 1 - base! 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!
A city's population is recorded every 5 years: Year: 0, 5, 10, 15 Population: 50,000; 53,000; 56,000; 59,000 Classify the pattern as exponential growth, exponential decay, or neither. (Hint: compare differences vs ratios.)
Answer: Neither; it shows approximately constant additive change (more linear than exponential). 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! Differences are +3000 each (constant, linear); ratios are 53000/50000=1.06, 56000/53000≈1.0566, 59000/56000≈1.0536—not constant, so not exponential. Choice C correctly identifies neither (more linear than exponential) through comparing constant differences vs. non-constant ratios. One distractor suggests exponential growth because it increases, but ratios aren't constant—nice try, but always verify both tests! 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!
A radioactive sample's mass is measured every day: Day: 0, 1, 2, 3 Mass (g): 80, 72, 64.8, 58.32 From the table, what is the percent rate of decay per day?
Answer: 10% decay per day (decay factor $b=0.90$ ) 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! Ratios: 72/80 = 0.9, 64.8/72 = 0.9, 58.32/64.8 = 0.9, constant at 0.9, so exponential decay with b=0.90, percent rate (1 - 0.9)*100% = 10% per day. Choice A correctly identifies 10% decay per day through constant ratios and the decay factor. A distractor like choice D could misread the factor as the percent directly, but subtract from 1 to get the rate—keep up the good work! 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!
A town's population is recorded every year. Year $y$ : 0, 1, 2, 3 Population $P(y)$ : 15000, 15600, 16224, 16872.96 Does the population show constant percent change? If so, what is the percent growth rate per year?
Answer: Yes; 4% growth per year 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! Let's calculate the ratios: 15600/15000 = 1.04, 16224/15600 = 1.04, 16872.96/16224 = 1.04. All ratios equal 1.04, confirming exponential growth with base 1.04. Since 1.04 = 1 + 0.04, the growth rate is 4% per year. Choice A correctly identifies 4% growth per year. Choice C incorrectly claims linear growth—while the first difference is 600, subsequent differences are 624, 648.96, which aren't constant, confirming exponential not linear growth. 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!
A laptop's value depreciates over time. The value (in dollars) is recorded each year: Year 0: 900 Year 1: 765 Year 2: 650.25 Year 3: 552.7125 Does this represent constant percent change? If so, is it growth or decay, and what is the percent rate per year?
Answer: Yes; exponential decay at 15% per year (ratio $0.85$ each year) 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! Let's calculate the ratios: Year 1/Year 0 = 765/900 = 0.85, Year 2/Year 1 = 650.25/765 = 0.85, Year 3/Year 2 = 552.7125/650.25 = 0.85. All ratios equal 0.85, confirming exponential decay with base 0.85. Choice A correctly identifies exponential decay at 15% per year through the constant ratio 0.85 (the laptop retains 85% of its value each year, losing 15%). Choice B incorrectly suggests growth when values are decreasing; Choice C wrongly claims it's linear (the differences are -135, -114.75, -97.5375, which are not constant); Choice D confuses the decay rate—if the ratio is 0.85, the laptop retains 85% and loses 15%, not decays at 85%. 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!
The scientist models the biomass recovery using $B(t) = 15(1.4)^t$ for the first phase, where $t$ is years after the fire and $B$ is in tons per hectare. Based on this model and the graph shown, what can be concluded about the ecosystem's recovery pattern?
Answer: The biomass increases by exactly 40% each year during the initial recovery phase, demonstrating exponential regrowth with consistent acceleration The model $B(t) = 15(1.4)^t$ shows exponential growth with factor 1.4, meaning 40% increase annually. This percent rate remains constant, though absolute amounts increase. Choice B misinterprets as linear growth. Choice C incorrectly suggests the percent rate changes. Choice D confuses the growth factor (1.4 = 140% of original) with growth rate (40% increase).
A store's price index is modeled by $I(t)=120(0.95)^t$ , where $t$ is in months. What percent does the index change per month, and is it growth or decay?
Answer: 5% decay per month 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! The model I(t)=120 $(0.95)^t$ has b=0.95 <1, indicating decay, with percent rate (1 - 0.95)*100% = 5% per month. Choice C correctly identifies 5% decay per month through the base in the exponential form. Something like choice B might confuse the remaining factor with the decay rate, but it's the complement— you're doing wonderfully! 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!
A laptop loses value each year due to depreciation. Its value is $1200 after 0 years, $1020 after 1 year, $867 after 2 years, and $736.95 after 3 years. From the data, what is the percent rate of change per year, and is it growth or decay?
Answer: 15% decay per year 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! Let's calculate the ratios: 1020/1200 = 0.85, 867/1020 = 0.85, 736.95/867 = 0.85. All ratios equal 0.85, confirming exponential decay with base 0.85. Choice B correctly identifies 15% decay per year through constant ratios: base 0.85 means each year retains 85% of previous value, losing 15% (100% - 85% = 15%). Choice C incorrectly claims 15% growth—but ratios less than 1 indicate decay, not growth! 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!
Which situation shows constant percent change (exponential) rather than constant additive change (linear)?
Answer: A population increases by 4% each year 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 situations, choice C describes a 4% increase each year, which means multiplying by 1.04 annually, a constant percent change. Choice C correctly identifies the exponential growth through the explicit percent language, distinguishing it from additive changes in the others. Distractors like choices A, B, and D use fixed amounts (3 liters, $50, 2 seconds), which are linear—excellent observation on the wording difference! 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!
A population study tracks two cities over time. City A has a population that can be modeled by $P_A(t) = 50000 \cdot 2^{t/10}$ , where $t$ is years since 2020. City B's population follows $P_B(t) = 45000(1.071)^t$ . After analyzing both models, which conclusion is most accurate?
Answer: City A doubles every 10 years with an annual growth rate of approximately 7.2%, while City B grows at exactly 7.1% per year City A's model $P_A(t) = 50000 \cdot 2^{t/10}$ shows the population doubles every 10 years. Converting to annual rate: $2^{1/10} \approx 1.072$ , so about 7.2% annually. City B grows at exactly 7.1% per year from the factor 1.071. Choice B misinterprets the exponent structure. Choice C incorrectly suggests City A's rate varies. Choice D misreads the doubling period as a growth rate.
A delivery company tracks total miles driven each day for a week. The sequence begins 120, 135, 150, 165 (miles). Is the change best described as exponential growth/decay or neither (linear/additive), and why?
Answer: Neither; it's linear because it adds 15 miles each day 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! Differences are constant: 135-120=15, 150-135=15, 165-150=15, indicating linear growth, while ratios (135/120=1.125, 150/135≈1.111, 165/150=1.1) are not constant, so it's neither exponential growth nor decay. Choice C correctly identifies neither, it's linear because it adds 15 miles each day, distinguishing from exponential via constant differences. A choice like A might check only one ratio and assume constancy, but always verify all ratios—super job being thorough! 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!
A lab culture is counted once per hour: Hour: 0, 1, 2, 3 Cells: 5,000; 5,750; 6,612.5; 7,604.375 Does this represent constant percent change? If yes, what is the percent growth rate per hour?
Answer: Yes; 15% growth per hour (growth factor $b=1.15$ ) 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! Ratios: 5,750/5,000 = 1.15, 6,612.5/5,750 = 1.15, 7,604.375/6,612.5 = 1.15, constant at 1.15, so yes, exponential growth with b=1.15, percent rate (1.15 - 1)*100% = 15% per hour. Choice A correctly identifies yes, 15% growth per hour through constant ratios and growth factor. A distractor like choice D might forget to multiply by 100 for percent, but remember to convert the decimal rate—fantastic effort! 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!
A radioactive sample has mass values (in grams) measured each day: Day: 0, 1, 2, 3 Mass (g): 200, 170, 144.5, 122.825 Does the data represent constant percent decay? If so, what is the percent decay rate per day?
Answer: Yes; 15% decay per day 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! Let's calculate the ratios: 170/200 = 0.85, 144.5/170 = 0.85, 122.825/144.5 = 0.85. All ratios equal 0.85, confirming exponential decay with base 0.85. Choice A correctly identifies 15% decay per day through constant ratios: base 0.85 means each day retains 85% of previous mass, losing 15% (100% - 85% = 15%). Choice B incorrectly states 85% decay, which would mean retaining only 15% each day (base 0.15), giving much faster decay! 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!
A radioactive isotope decays such that 12.5% of the original amount remains after 30 days. A chemist needs to determine the daily decay rate to predict when only 1% will remain. What is the constant daily percent decay rate?
Answer: The isotope decays at approximately 6.9% per day since $0.125 = (0.931)^{30}$ and $1 - 0.931 = 0.069$ When you encounter radioactive decay problems, remember that decay follows an exponential model, not a linear one. The amount remaining follows the formula $A = A_0(1-r)^t$ , where $r$ is the daily decay rate. To find the daily decay rate, you need to work backwards from the given information. You know that 12.5% (or 0.125) remains after 30 days, so: $0.125 = (1-r)^{30}$ Taking the 30th root of both sides: $(1-r) = (0.125)^{1/30} = 0.931$ Therefore: $r = 1 - 0.931 = 0.069$ or about 6.9% daily decay. Answer A incorrectly assumes the half-life is 10 days and uses an inappropriate formula. The actual half-life would be much shorter given that only 12.5% remains after 30 days. Answer B makes the critical error of assuming linear decay, simply dividing the total percentage lost by the number of days. This ignores the exponential nature of radioactive decay. Answer C commits the same linear thinking mistake as B, treating decay as if the same absolute amount is lost each day rather than the same percentage of what remains. The key study tip for exponential decay problems: never use linear division to find rates. Always use the exponential formula $(1-r)^t = \text{fraction remaining}$ and solve for $r$ by taking the appropriate root. Radioactive decay is inherently exponential because each day you lose a percentage of what's left, not a fixed amount.
A bacteria culture is measured every hour. The number of bacteria is shown below. Does this represent constant percent change (exponential) or constant additive change (linear), and what is the percent rate per hour? Time (hours): 0, 1, 2, 3 Bacteria count: 200, 240, 288, 345.6
Answer: Exponential growth at 20% per hour (growth factor $b=1.2$ ) 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! Let's calculate the ratios: 240/200 = 1.2, 288/240 = 1.2, 345.6/288 = 1.2. All ratios equal 1.2, confirming exponential growth with base b = 1.2. Since 1.2 = 1 + 0.2, this represents 20% growth per hour. Choice B correctly identifies exponential growth at 20% per hour (growth factor b=1.2) through the constant ratio of 1.2. Choice A incorrectly suggests linear growth, but the differences aren't constant (240-200=40, but 288-240=48, not 40). 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!
A population is measured every year: Year 0: 40,000 Year 1: 41,600 Year 2: 43,264 Year 3: 44,994.56 Classify the pattern as exponential growth, exponential decay, or neither, and identify the percent rate per year.
Answer: Exponential growth at 4% per year 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! Let's calculate the ratios: 41,600/40,000 = 1.04, 43,264/41,600 = 1.04, 44,994.56/43,264 = 1.04. All ratios equal 1.04, confirming exponential growth with base 1.04. To find the percent rate: 1.04 - 1 = 0.04 = 4% growth per year. Choice A correctly identifies exponential growth at 4% per year through the constant ratio 1.04 (representing 104% of previous value). Choice B has the wrong direction (decay instead of growth); Choice C incorrectly claims it's linear based on the first difference only (differences are 1,600, 1,664, 1,730.56—not constant); Choice D confuses the percent notation. 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!
A laptop's value depreciates each year. The values are: Year 0: $1200 Year 1: $1020 Year 2: $867 Year 3: $736.95 Is this constant percent growth or decay, and what is the percent rate per year?
Answer: Exponential decay at 15% per year. 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! To check, compute the ratios from the table: 1020/1200 = 0.85, 867/1020 = 0.85, and 736.95/867 = 0.85, showing a constant ratio of 0.85, which fits the exponential form V(t) = 1200 * $(0.85)^t$ . Choice B correctly identifies exponential decay at 15% per year through constant ratios of 0.85 (since 1 - 0.85 = 0.15 or 15%). One distractor suggests linear decay by $180, but differences are 1200-1020=180, 1020-867=153, 867-736.95=130.05—not constant, so it's not linear; remember to check ratios for exponential patterns! 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! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!

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