Algebra 2 Quiz: Recognize Percent Growth Or Decay
20 questions · exam conditions
0:00
Recognize Percent Growth Or DecayQuestion 1 of 20

A company tracks the number of active users each month:

Month: 0, 1, 2, 3 Users: 12,000; 13,200; 14,520; 15,972

Does this represent constant percent change, and if so, what is the percent growth rate per month?

No; it's linear because it increases by 1,200 each month
Yes; 12% growth per month (growth factor b=1.12b=1.12)
Yes; 10% growth per month (growth factor b=1.10b=1.10)
Yes; 1.1% growth per month (because ratios are 1.1)
← Back to quizzes

Algebra 2 Quiz

Algebra 2 Quiz: Recognize Percent Growth Or Decay

Practice Recognize Percent Growth Or Decay in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Recognize Percent Growth Or Decay, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A company tracks the number of active users each month:

Month: 0, 1, 2, 3 Users: 12,000; 13,200; 14,520; 15,972

Does this represent constant percent change, and if so, what is the percent growth rate per month?

  1. No; it's linear because it increases by 1,200 each month
  2. Yes; 12% growth per month (growth factor b=1.12b=1.12)
  3. Yes; 10% growth per month (growth factor b=1.10b=1.10) (correct answer)
  4. Yes; 1.1% growth per month (because ratios are 1.1)
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! Applying the ratio test: 13,200/12,000 = 1.1, 14,520/13,200 = 1.1, 15,972/14,520 = 1.1, so constant ratios at 1.1 confirm exponential growth with b=1.10, and percent rate (1.10 - 1)*100% = 10% per month. Choice C correctly identifies yes, 10% growth per month through these constant ratios and rate calculation. A distractor like choice A might confuse it with linear due to increasing values, but the differences (1,200; 1,320; 1,452) aren't constant, so it's not linear—nice 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)!

Question 2

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?

  1. 10% growth per year
  2. 15% decay per year (correct answer)
  3. 15% growth per year
  4. 0.85% decay 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! 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)!

Question 3

A recycling bin contains 500 pounds of material. Each day, 6% of the material is removed. Which statement correctly identifies the constant percent rate and the exponential model for the amount remaining after dd days?

  1. 6% decay; M(d)=500(0.94)dM(d)=500(0.94)^d (correct answer)
  2. 94% decay; M(d)=500(0.06)dM(d)=500(0.06)^d
  3. 6% growth; M(d)=500(1.06)dM(d)=500(1.06)^d
  4. Linear decay; M(d)=500−0.06dM(d)=500-0.06d
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! If 6% is removed each day, then 94% remains (100% - 6% = 94%). This means we multiply by 0.94 each day: Day 0: 500 pounds, Day 1: 500 × 0.94, Day 2: 500 × (0.94)², giving us M(d) = 500(0.94)^d. Choice B correctly identifies 6% decay with the model M(d) = 500(0.94)^d (base 0.94 represents 94% remaining). Choice A incorrectly suggests growth when material is being removed; Choice C misinterprets the situation as 94% decay (which would leave only 6%); Choice D suggests a linear model which doesn't match percent removal. 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)!

Question 4

Two savings plans are described below.

Plan 1: Start with $2000 and add $50 each month. Plan 2: Start with $2000 and increase the balance by 2% each month.

Which plan shows constant percent change (exponential), and which shows constant additive change (linear)?

  1. Plan 1 is exponential; Plan 2 is linear
  2. Both plans are exponential because both increase each month
  3. Both plans are linear because they both change monthly
  4. Plan 2 is exponential; Plan 1 is linear (correct answer)
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! Plan 1 adds $50 each month—this is constant additive change, making it linear. Plan 2 increases the balance by 2% each month, meaning each month's balance is 102% of the previous (multiply by 1.02)—this is constant percent change, making it exponential. Choice D correctly identifies Plan 2 as exponential (constant percent change) and Plan 1 as linear (constant additive change). Choice A reverses these classifications, missing that "add $50" signals linear while "increase by 2%" signals 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!

Question 5

Two phone plans track the number of users over months. Plan A adds 250 users each month. Plan B increases the number of users by 5% each month. Which statement is correct about the type of change for each plan?

  1. Plan A is exponential; Plan B is linear
  2. Both plans are linear because they change each month
  3. Plan A is linear (constant additive); Plan B is exponential (constant percent) (correct answer)
  4. Both plans are exponential because they both 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! Plan A adds 250 users each month—this is a constant amount added, making it linear (if starting at N₀, then N(t) = N₀ + 250t). Plan B increases by 5% each month—this is a constant percent change, making it exponential (if starting at N₀, then N(t) = N₀(1.05)^t). Choice C correctly identifies Plan A as linear (constant additive change of +250) and Plan B as exponential (constant percent change of 5%). Choice A reverses the classifications; Choice B incorrectly claims both are linear; Choice D incorrectly claims both are exponential just because they increase. 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)!

Question 6

A scientist records a sample's mass (in grams) after each processing step:

Step nn: 0, 1, 2, 3 Mass M(n)M(n): 200, 180, 162, 145.8

Classify the pattern and identify the constant percent rate per step.

  1. Neither; it is linear because it decreases by 20 grams each step.
  2. Exponential growth at 10% per step.
  3. Exponential decay at 0.90% per step.
  4. Exponential decay at 10% per step. (correct answer)
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)!

Question 7

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.)

  1. Exponential growth, because the ratio is constant.
  2. Exponential decay, because the population increases by smaller amounts each interval.
  3. Neither; it shows approximately constant additive change (more linear than exponential). (correct answer)
  4. Exponential growth at 6% per 5 years.
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)!

Question 8

A radioactive sample's mass is recorded every day:

Day: 0, 1, 2, 3 Mass (g): 50, 46, 42, 38

Does this represent constant percent decay? (Hint: constant percent change means a constant ratio, not a constant difference.)

  1. Yes; it is exponential decay because it decreases by 4 g each day
  2. Yes; 4% decay per day because it decreases by 4 each day
  3. Yes; 8% decay per day because 46/50=0.9246/50=0.92
  4. No; the ratios 46/5046/50, 42/4642/46, and 38/4238/42 are not all equal (correct answer)
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! Let's check the ratios: 46/50 = 0.92, 42/46 ≈ 0.913, 38/42 ≈ 0.905. The ratios are not constant (0.92 ≠ 0.913 ≠ 0.905), so this is NOT exponential decay. Instead, the differences are constant: 50-46=4, 46-42=4, 42-38=4, indicating linear decay. Choice B correctly identifies that the ratios are not all equal, confirming this is not constant percent decay. Choice C incorrectly claims exponential decay based on the constant difference of 4g, which actually indicates linear change. 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!

Question 9

An investment account follows the model A(t)=2500(1.06)tA(t)=2500(1.06)^t, where tt is in years.

Identify the constant percent rate and classify the change.

  1. Exponential decay at 6% per year
  2. Linear growth of 0.06⋅2500=1500.06\cdot 2500=150 dollars per year
  3. Exponential growth at 6% per year (correct answer)
  4. Exponential growth at 1.06% 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! From the formula A(t) = 2500(1.06)^t, we identify the base as 1.06. Since 1.06 > 1, this represents exponential growth. The percent rate is calculated as: rate = base - 1 = 1.06 - 1 = 0.06 = 6%. Choice C correctly identifies exponential growth at 6% per year through base analysis: base 1.06 means multiplying by 1.06 each year, which is 106% of previous value, representing 6% growth. Choice D incorrectly states 1.06% growth—confusing the base 1.06 with the percent rate; the base 1.06 corresponds to 6% growth, not 1.06% 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)!

Question 10

A gym membership fee changes over time according to the table.

Month mm: 0, 1, 2, 3 Fee F(m)F(m): 40, 44, 48, 52

Classify the change as exponential growth, exponential decay, or neither.​​

  1. Exponential growth because the fee increases each month
  2. Neither; it shows constant additive change (linear) (correct answer)
  3. Exponential decay because the ratios are less than 1
  4. Exponential growth at 10% per 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! Let's check the differences: 44-40 = 4, 48-44 = 4, 52-48 = 4. The differences are constant at +4 per month, indicating linear change. Let's verify with ratios: 44/40 = 1.1, 48/44 ≈ 1.091, 52/48 ≈ 1.083. The ratios are not constant, confirming this is not exponential. Choice B correctly identifies this as neither exponential growth nor decay, but rather constant additive (linear) change. Choice A incorrectly claims exponential growth just because values increase, while choice D wrongly calculates a 10% growth rate when the ratios aren't even constant. 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!

Question 11

A savings account is modeled by the function A(t)=2500(1.04)tA(t)=2500(1.04)^t, where tt is in years.

What is the constant percent rate of change, and is it growth or decay?

  1. 1.04% growth per year.
  2. It is linear because 25002500 is the starting amount.
  3. 4% decay per year.
  4. 4% growth per year. (correct answer)
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)!

Question 12

A radioactive sample has mass values recorded each day.

Day dd: 0, 1, 2, 3 Mass m(d)m(d) (g): 200, 170, 144.5, 122.825

From the table, determine whether the sample shows constant percent change. If it does, find the percent decay rate per day.​​

  1. Yes; 15% decay per day (correct answer)
  2. Yes; 17% decay per day
  3. Yes; 15% growth per day
  4. No; it is linear because it decreases by 30 g 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! 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. Since 0.85 = 1 - 0.15, the decay rate is 15% per day (the sample retains 85% of its mass each day). Choice A correctly identifies 15% decay per day. Choice D incorrectly claims linear decay—the differences are 30, 25.5, 21.675, which aren't constant, while the ratios are constant at 0.85, confirming exponential 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!

Question 13

A medication amount in the bloodstream is modeled by M(t)=80(0.92)tM(t)=80(0.92)^t, where tt is in hours.

What percent does the amount change each hour, and is it growth or decay?​

  1. 8% decay per hour (correct answer)
  2. 92% decay per hour
  3. 8% growth per hour
  4. 0.92% decay per 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! 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)!

Question 14

A savings account balance follows the function B(t)=1500(1.03)tB(t)=1500(1.03)^t, where tt is in years. What is the constant percent rate of change, and is it growth or decay?

  1. 3% growth per year (correct answer)
  2. 1.03% growth per year
  3. 3% decay per year
  4. 103% growth 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! From the function B(t) = 1500(1.03)^t, the base is 1.03. Since 1.03 > 1, this represents growth. To find the percent rate: 1.03 - 1 = 0.03 = 3% growth per year. Choice A correctly identifies 3% growth per year through the base 1.03 analysis (103% of previous value = 3% growth). Choice B incorrectly states 1.03% instead of 3%; Choice C has the wrong direction (decay instead of growth); Choice D misinterprets the base as the percent rate itself. 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)!

Question 15

A quantity yy is recorded for integer values of xx:

xx: 1, 2, 3, 4 yy: 30, 36, 43.2, 51.84

From the table, what is the percent growth rate per 1-unit increase in xx?

  1. 20% growth (correct answer)
  2. 6% growth
  3. 12% growth
  4. 20% decay
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! Let's calculate the ratios: 36/30 = 1.2, 43.2/36 = 1.2, 51.84/43.2 = 1.2. All ratios equal 1.2, confirming exponential growth with base 1.2. Choice A correctly identifies 20% growth through constant ratios: base 1.2 means multiplying by 1.2 each unit increase in x, which is 120% of previous value, representing 20% growth (120% - 100% = 20%). Choice C incorrectly states 12% growth, which would give base 1.12 and different values! 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)!

Question 16

Which situation shows constant percent change (exponential) rather than constant additive change (linear)?​

  1. A population increases by 4% each year (correct answer)
  2. A tank is filled by adding 3 liters every minute
  3. A salary increases by $50 each month
  4. A runner improves time by 2 seconds each week
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 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)!

Question 17

A town's population is 18,000 and grows by 4% each year.

Which function models the population after tt years (with t=0t=0 now)?

  1. P(t)=18000+0.04tP(t)=18000+0.04t
  2. P(t)=18000(0.96)tP(t)=18000(0.96)^t
  3. P(t)=18000(1.04)tP(t)=18000(1.04)^t (correct answer)
  4. P(t)=18000(4)tP(t)=18000(4)^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! The problem states the population grows by 4% each year, which means exponential growth. For 4% growth, the base is 1 + 0.04 = 1.04. Starting at 18,000, the exponential model is P(t) = 18000(1.04)^t. Choice C correctly identifies the exponential model P(t) = 18000(1.04)^t through proper base calculation: 4% growth means multiplying by 1.04 each year. Choice A incorrectly uses a linear model 18000 + 0.04t, which would add only 0.04 people per year—clearly wrong for percent 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)!

Question 18

A medication amount in the bloodstream is modeled by M(t)=80(0.92)tM(t)=80(0.92)^t, where tt is in hours.

What percent does the amount change each hour, and is it growth or decay?

  1. 8% decay per hour (correct answer)
  2. 92% decay per hour
  3. 8% growth per hour
  4. 0.92% decay per 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! 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)!

Question 19

A store's price index is modeled by I(t)=120(0.95)tI(t)=120(0.95)^t, where tt is in months. What percent does the index change per month, and is it growth or decay?

  1. 5% decay per month (correct answer)
  2. 0.95% decay per month
  3. 95% decay per month
  4. 5% growth per 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! 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)!

Question 20

A medicine dose in the bloodstream decreases by 12% each hour. Which exponential function models the amount remaining after tt hours if the initial amount is 80 mg?

  1. A(t)=80(0.12)tA(t)=80(0.12)^t
  2. A(t)=80−0.12tA(t)=80-0.12t
  3. A(t)=80(1.12)tA(t)=80(1.12)^t
  4. A(t)=80(0.88)tA(t)=80(0.88)^t (correct answer)
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! If the medicine decreases by 12% each hour, then 88% remains (100% - 12% = 88%). This means we multiply by 0.88 each hour: Hour 0: 80 mg, Hour 1: 80 × 0.88, Hour 2: 80 × (0.88)², giving us A(t) = 80(0.88)^t. Choice C correctly identifies the exponential decay model with base 0.88 (representing 88% remaining after each hour). Choice A incorrectly uses base 1.12 (which would be 12% growth, not decay); Choice B wrongly uses 0.12 as the base (this would mean only 12% remains, not that 12% is lost); Choice D suggests a linear model which doesn't match percent 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)!