Recognize Percent Growth or Decay
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Algebra 2 › Recognize Percent Growth or Decay
A scientist records a sample’s mass (in grams) after each processing step:
Step $n$: 0, 1, 2, 3
Mass $M(n)$: 200, 180, 162, 145.8
Classify the pattern and identify the constant percent rate per step.
Exponential decay at 10% per step.
Exponential growth at 10% per step.
Neither; it is linear because it decreases by 20 grams each step.
Exponential decay at 0.90% per step.
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!
To check, compute the ratios from the table: 180/200 = 0.9, 162/180 = 0.9, and 145.8/162 = 0.9, showing a constant ratio of 0.9, fitting exponential decay.
Choice C correctly identifies exponential decay at 10% per step through constant ratios of 0.9 (since 1 - 0.9 = 0.1 or 10%).
One distractor suggests linear by 20 grams, but differences are 20, 18, 16.2—not constant; excellent work spotting the ratio pattern instead!
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 savings account is modeled by the function $A(t)=2500(1.04)^t$, where $t$ is in years.
What is the constant percent rate of change, and is it growth or decay?
4% decay per year.
4% growth per year.
1.04% growth per year.
It is linear because $2500$ is the starting amount.
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!
The function A(t) = 2500(1.04)^t is already in exponential form, with base b=1.04, indicating a constant multiplicative factor of 1.04 per year.
Choice A correctly identifies 4% growth per year through base analysis (since 1.04 - 1 = 0.04 or 4%).
One distractor confuses the rate with 1.04%, but the percent rate is (b - 1) * 100%, so 0.04 is 4%, not 1.04%—keep practicing to spot this 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 quantity is modeled by $y=600(0.97)^t$, where $t$ is measured in days.
Which statement correctly describes the situation?
Exponential decay at 3% per day.
Exponential decay at 0.97% per day.
Linear decay at 3 units per day.
Exponential growth at 3% per day.
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!
The function y = 600(0.97)^t is in exponential form, with base b=0.97, indicating a constant multiplicative factor of 0.97 per day.
Choice B correctly identifies exponential decay at 3% per day through base analysis (since 1 - 0.97 = 0.03 or 3%).
One distractor suggests linear decay, but exponential functions like this have varying differences, not constant additions—keep practicing to recognize the form!
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 streaming service tracks subscribers each month:
Month $m$: 1, 2, 3, 4
Subscribers: 12,000; 12,600; 13,230; 13,891.5
Does the table show a constant percent growth rate? If yes, what is the percent growth rate per month?
Yes; 105% growth per month (because the factor is $1.05$).
Yes; 6% growth per month (growth factor $b=1.06$).
Yes; 5% growth per month (growth factor $b=1.05$).
No; it is linear because it increases by about 600 subscribers each month.
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!
To check, compute the ratios from the table: 12600/12000 = 1.05, 13230/12600 = 1.05, and 13891.5/13230 = 1.05, showing a constant ratio of 1.05, fitting exponential growth.
Choice B correctly identifies yes, 5% growth per month through constant ratios of 1.05 (since 1.05 - 1 = 0.05 or 5%).
One distractor suggests linear by about 600, but differences are 600, 630, 661.5—not constant; you're on the right track checking ratios first!
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 phone battery retains a constant percent of its charge every hour. The charge levels are:
Hour $t$: 0, 1, 2, 3
Charge (%): 100, 92, 84.64, 77.8688
What percent does the battery decrease per hour?
92% decay per hour.
8% decay per hour.
8% growth per hour.
Neither; it is linear because it drops by 8% each hour.
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!
To check, compute the ratios from the table: 92/100 = 0.92, 84.64/92 = 0.92, and 77.8688/84.64 = 0.92, showing a constant ratio of 0.92, fitting exponential decay.
Choice A correctly identifies 8% decay per hour through constant ratios of 0.92 (since 1 - 0.92 = 0.08 or 8%).
One distractor suggests 92% decay, but that's confusing the factor with the rate—the rate is the change, so 8%; you're doing great learning this distinction!
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 medication amount remaining in the bloodstream is recorded every hour:
Hour $t$: 0, 1, 2, 3
Amount (mg): 60, 51, 43.35, 36.8475
From the table, what is the percent rate of change per hour, and is it growth or decay?
Neither; it is linear because it decreases by 9 mg each hour.
Growth at 15% per hour (growth factor $b=1.15$).
Decay at 15% per hour (decay factor $b=0.85$).
Decay at 10% per hour (decay factor $b=0.90$).
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!
To check, compute the ratios from the table: 51/60 = 0.85, 43.35/51 = 0.85, and 36.8475/43.35 = 0.85, showing a constant ratio of 0.85, which fits the exponential decay form.
Choice A correctly identifies decay at 15% per hour through constant ratios of 0.85 (since 1 - 0.85 = 0.15 or 15%).
One distractor suggests linear decay by 9 mg, but differences are 60-51=9, 51-43.35=7.65, 43.35-36.8475=6.5025—not constant, so it's exponential; great job checking both!
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)!
Two gym membership plans track total cost after $m$ months:
Plan P: $25, 50, 75, 100$ (for $m=1,2,3,4$)
Plan Q: $25, 27.50, 30.25, 33.275$ (for $m=1,2,3,4$)
Which plan shows constant percent change (exponential), and which shows constant additive change (linear)?
Both are exponential because both increase each month.
Plan P is exponential (constant percent); Plan Q is linear (constant additive).
Neither is exponential because neither uses an exponent in the table.
Plan Q is exponential (constant percent); Plan P is linear (constant additive).
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!
For Plan P, differences are +25 each (linear); for Plan Q, ratios are 27.50/25=1.1, 30.25/27.50=1.1, 33.275/30.25=1.1, constant at 1.1, so Q is exponential growth.
Choice B correctly identifies Plan Q as exponential (constant percent) and Plan P as linear (constant additive) through ratio and difference checks.
One distractor says both are exponential since both increase, but only constant ratios make it exponential—keep up the good work distinguishing them!
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?
Exponential growth at 15% per year.
Exponential decay at 0.85% per year (since $0.85$ is in the table).
Linear decay because it decreases by $\$180$ each year.
Exponential decay at 15% per year.
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!
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)!
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.)
Exponential growth at 6% per 5 years.
Exponential decay, because the population increases by smaller amounts each interval.
Neither; it shows approximately constant additive change (more linear than exponential).
Exponential growth, because the ratio is constant.
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!
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 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?
Exponential growth, because the ratio is constant: $135/120=1.125$
Exponential decay, because the differences are constant
Neither; it’s linear because it adds 15 miles each day
Neither; it must be quadratic because the numbers increase
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! 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)!