This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. The key difference: linear growth adds the same amount each time (constant rate: +50, +50, +50), while exponential growth multiplies by the same percent each time (constant ratio: ×1.1, ×1.1, ×1.1). To check which: if differences are constant, it's linear; if ratios are constant, it's exponential. Example: 100, 150, 200, 250 has constant differences (+50) = linear. But 100, 110, 121, 133.1 has constant ratios (×1.1) = exponential! Let's contrast: Options B, C, and D all describe adding the same amount each time—constant additive change. But option A describes earning 2% interest each month, which involves multiplying by the same percent each time—constant multiplicative change. The exponential case has the account balance multiplied by 1.02 each month, while the linear cases would have constant differences like +5 liters, +1 mile, or +$1. This is exponential growth! Choice A correctly identifies the bank account earning 2% interest as exponential because the balance is multiplied by 1.02 each month (constant percent change), not increased by a fixed dollar amount. Choices B, C, and D all describe linear situations: they involve adding a constant amount (5 liters, 1 mile, $1) rather than multiplying by a constant factor. Adding the same amount = linear, multiplying by the same percent = exponential! Context language decoder for exponential: 'grows by X% per year,' 'decreases by X% per month,' 'X% interest compounded,' 'doubles every,' 'halves every,' 'increases X-fold' → all signal constant percent change (exponential). But 'adds $X per period' or 'increases by X units' → constant additive change (linear). The 'percent per period' pattern is the key giveaway!

This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent decay means the quantity is multiplied by the same factor between 0 and 1 each time period: if a car depreciates by 15% per year, it's multiplied by 0.85 each year (since keeping 85% = 1 - 0.15 = 0.85 means 'you lose 15%'). The quantity shrinks exponentially, approaching but never quite reaching zero. Looking at the function $P(t)=12,000·(0.98)^t$: the base 0.98 is less than 1, indicating exponential decay. The percent rate is calculated from r = 0.98 - 1 = -0.02 = 2% decay. This means each time t increases by 1, P is multiplied by 0.98, which is a 2% decrease. Choice B correctly identifies this as exponential decay at 2% per year because the base <1 and |r|=0.02 confirms the rate. Choice D has the percent rate wrong: the base 0.98 doesn't mean 98% decay. When b = 0.98, we subtract 1 to get the rate: 0.98 - 1 = -0.02 = 2% decay. The base includes the remaining 98% (the '0.98'), so the decay is 2%, not 98%! To find percent rate from a growth/decay factor: (1) Identify b (the base or factor), (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: b = 1.12 → r = 0.12 → 12% growth. For decay: b = 0.95 → r = -0.05 → 5% decay (we usually state as positive '5% decay' rather than 'negative 5%'). The subtraction of 1 is the crucial step!

This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent decay means the quantity is multiplied by the same factor between 0 and 1 each time period: if a car depreciates by 15% per year, it's multiplied by 0.85 each year (since keeping 85% = 1 - 0.15 = 0.85 means 'you lose 15%'). The quantity shrinks exponentially, approaching but never quite reaching zero. The context describes 'a car worth $20,000 depreciates by 15% each year.' Key phrase: 'depreciates by 15%' directly tells us this is exponential decay with a constant percent rate. Each year, the quantity is multiplied by 1 - 0.15 = 0.85, making this exponential rather than linear. In one year, you have 0.85 × 100% of what you started with (original 100% minus 15%). Choice B correctly identifies this as exponential decay at 15% per year because the context percent language shows constant multiplicative change by 0.85. Choice A confuses exponential with linear: it sees the pattern of decreasing values and assumes linear, but we need to check how they're changing. Showing that differences are not constant (e.g., first year -3000 from 20k, next -2550 from 17k) while ratios are constant (×0.85) means exponential! Constant addition = linear, constant multiplication = exponential! Context language decoder for exponential: 'grows by X% per year,' 'decreases by X% per month,' 'X% interest compounded,' 'doubles every,' 'halves every,' 'increases X-fold' → all signal constant percent change (exponential). But 'adds $X per period' or 'increases by X units' → constant additive change (linear). The 'percent per period' pattern is the key giveaway! Real-world clue: exponential growth/decay contexts involve processes where the amount of change depends on how much you currently have: 'Population grows by 5% per year' means a population of 1000 gains 50, but a population of 10,000 gains 500—the change is bigger when the base is bigger. That's exponential! Linear is when you add the same amount regardless of current size.

This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent growth means the quantity is multiplied by the same factor greater than 1 each time period: if a population grows by 2% per year, it's multiplied by 1.02 each year (since 102% = 1 + 0.02 = 1.02 means 'keep all of what you had plus gain 2% more'). This creates exponential growth where the amount added each period gets larger because you're taking a percent of an increasing base! Looking at the function P(t) = $12000(1.02)^t$: the base 1.02 is greater than 1, indicating exponential growth. The percent rate is calculated from r = 1.02 - 1 = 0.02 = 2% growth. This means each time t increases by 1, P is multiplied by 1.02, which is a 2% increase. Choice C correctly identifies this as exponential growth: increases 2% per year because the base 1.02 means the population grows by 2% annually. Choice D has the percent rate wrong: the base 1.02 doesn't mean 102% growth. When b = 1.02, we subtract 1 to get the rate: 1.02 - 1 = 0.02 = 2% growth. The base includes the original 100% (the '1') plus the growth rate (the '0.02'), so it's 2%, not 102%! To find percent rate from a growth/decay factor: (1) Identify b (the base or factor), (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: b = 1.02 → r = 0.02 → 2% growth. The subtraction of 1 is the crucial step!

This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent decay means the quantity is multiplied by the same factor between 0 and 1 each time period: if a medication decreases by 20% per hour, it's multiplied by 0.80 each hour (since keeping 80% = 1 - 0.20 = 0.80 means 'you lose 20%'). The quantity shrinks exponentially, approaching but never quite reaching zero. The context describes 'decreases by 20% every hour.' Key phrase: 'decreases by 20%' directly tells us this is exponential decay with a constant percent rate. Each hour, the quantity is multiplied by 1 - 0.20 = 0.80, making this exponential rather than linear. In one hour, you have 80% of what you started with (original 100% minus 20%). Choice B correctly identifies this as A(t) = 80(0.80)^t because decreasing by 20% means multiplying by 0.80 each hour, and the initial amount is 80 mg. Choice A says growth when it's actually decay: looking at the context, since it describes 'decreases,' this is decay, not growth. When the context says 'decreases,' 'depreciates,' or 'decays,' that's exponential decay with a base between 0 and 1! Context language decoder for exponential: 'grows by X% per year,' 'decreases by X% per month,' 'X% interest compounded,' 'doubles every,' 'halves every,' 'increases X-fold' → all signal constant percent change (exponential). But 'adds $X per period' or 'increases by X units' → constant additive change (linear). The 'percent per period' pattern is the key giveaway!

This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. From a table, to identify exponential with constant percent rate: divide consecutive y-values to find ratios. If y₂/y₁ = y₃/y₂ = y₄/y₃ = same number, that's your growth/decay factor b. If b > 1 (like 1.08), it's growth at (b-1)×100% = 8%. If 0 < b < 1 (like 0.92), it's decay at (1-b)×100% = 8% decay. Let's check if this is exponential by finding ratios: From year 0 to 1: 21,600/20,000 = 1.08. From year 1 to 2: 23,328/21,600 = 1.08. From year 2 to 3: 25,194.24/23,328 = 1.08. All ratios equal 1.08, confirming exponential! Since 1.08 is greater than 1, this is growth. The percent rate is 1.08 - 1 = 0.08 = 8%. Choice A correctly identifies this as exponential growth at 8% per year because all consecutive ratios equal 1.08 and 1.08 - 1 = 0.08 = 8% growth. Choice C confuses exponential with linear: it sees the pattern of increasing values and calculates the difference 21,600 - 20,000 = 1,600, but we need to check how they're changing. The differences are NOT constant (21,600 - 20,000 = 1,600; 23,328 - 21,600 = 1,728; 25,194.24 - 23,328 = 1,866.24), but the ratios ARE constant (all 1.08). Constant addition = linear, constant multiplication = exponential! The ratio test for exponential from a table: (1) Divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, etc., (2) If all ratios are equal, it's exponential and that ratio is your growth/decay factor b, (3) If b > 1, it's growth; if 0 < b < 1, it's decay, (4) Calculate percent rate: r = b - 1, convert to percent. Example: ratios all equal 1.06 → exponential growth, 6% per period. Easy!

This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent growth means the quantity is multiplied by the same factor greater than 1 each time period: if a population grows by 5% per year, it's multiplied by 1.05 each year (since 105% = 1 + 0.05 = 1.05 means 'keep all of what you had plus gain 5% more'). This creates exponential growth where the amount added each period gets larger because you're taking a percent of an increasing base! The context describes 'a bacteria culture starts with 1000 cells and increases by 20% each hour.' Key phrase: 'increases by 20% each hour' directly tells us this is exponential growth with a constant percent rate. Each hour, the quantity is multiplied by 1 + 0.20 = 1.20, making this exponential rather than linear. In one hour, you have 120% of what you started with (original 100% plus 20%). Choice B correctly identifies this as b=1.20 and r=0.20 (20% growth) because the growth factor includes the original 100% plus the 20% increase. Choice A says growth when it's actually decay (or vice versa): looking at the context, since it describes increase, this is growth, not decay. When the base is greater than 1, or when the context says 'increases,' that's exponential growth! To find percent rate from a growth/decay factor: (1) Identify b (the base or factor), (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: b = 1.12 → r = 0.12 → 12% growth. For decay: b = 0.95 → r = -0.05 → 5% decay (we usually state as positive '5% decay' rather than 'negative 5%'). The subtraction of 1 is the crucial step! Context language decoder for exponential: 'grows by X% per year,' 'decreases by X% per month,' 'X% interest compounded,' 'doubles every,' 'halves every,' 'increases X-fold' → all signal constant percent change (exponential). But 'adds $X per period' or 'increases by X units' → constant additive change (linear). The 'percent per period' pattern is the key giveaway!

This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. From a table, to identify exponential with constant percent rate: divide consecutive y-values to find ratios. If y₂/y₁ = y₃/y₂ = y₄/y₃ = same number, that's your growth/decay factor b. If b > 1 (like 1.08), it's growth at (b-1)×100% = 8%. If 0 < b < 1 (like 0.92), it's decay at (1-b)×100% = 8% decay. Let's check if this is exponential by finding ratios: From year 0 to 1: 9,000/10,000 = 0.90. From year 1 to 2: 8,100/9,000 = 0.90. From year 2 to 3: 7,290/8,100 = 0.90. All ratios equal 0.90, confirming exponential! Since 0.90 is less than 1, this is decay. The percent rate is 0.90 - 1 = -0.10 = 10% decay. Choice C correctly identifies this as exponential decay at 10% per year because all consecutive ratios equal 0.90, meaning the population is multiplied by 0.90 each year—it retains 90% and loses 10%. Choice D says decay when it's actually decay (or vice versa): looking at the base 0.90, since 0.90 < 1, this is decay, not growth. When the base is between 0 and 1, or when the context shows decreasing values, that's exponential decay! But choice D also mistakes 0.90 for meaning 90% decay—the decay rate is 10%, not 90%! The ratio test for exponential from a table: (1) Divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, etc., (2) If all ratios are equal, it's exponential and that ratio is your growth/decay factor b, (3) If b > 1, it's growth; if 0 < b < 1, it's decay, (4) Calculate percent rate: r = b - 1, convert to percent. Example: ratios all equal 1.06 → exponential growth, 6% per period. Easy!

This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. The key difference: linear growth adds the same amount each time (constant rate: +50, +50, +50), while exponential growth multiplies by the same percent each time (constant ratio: ×1.1, ×1.1, ×1.1). To check which: if differences are constant, it's linear; if ratios are constant, it's exponential. Example: 100, 150, 200, 250 has constant differences (+50) = linear. But 100, 110, 121, 133.1 has constant ratios (×1.1) = exponential! Let's contrast: (a) gym membership costs $40 per month involves adding the same amount each time—constant additive change. (b) car's value decreases by 10% each year involves multiplying by the same percent each time—constant multiplicative change. (c) water tank filled at 3 gallons per minute is adding 3 gallons each time. (d) plant grows 2 cm each week is adding 2 cm each time. Only (b) has constant ratios, while the others have constant differences. This is exponential decay! Choice A correctly identifies only (b) as involving constant percent rate because 'decreases by 10% each year' means the car retains 90% of its value each year—multiply by 0.90, which is exponential decay. Choice C sees 'percent' in option (b) and correctly identifies it as exponential, but mistakenly includes (d) which says 'grows 2 cm each week'—that's adding the same length each time, not multiplying by the same percent. Growing by a fixed amount is linear, not exponential! Context language decoder for exponential: 'grows by X% per year,' 'decreases by X% per month,' 'X% interest compounded,' 'doubles every,' 'halves every,' 'increases X-fold' → all signal constant percent change (exponential). But 'adds $X per period' or 'increases by X units' → constant additive change (linear). The 'percent per period' pattern is the key giveaway!

This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. The key difference: linear growth adds the same amount each time (constant rate: +50, +50, +50), while exponential growth multiplies by the same percent each time (constant ratio: ×1.1, ×1.1, ×1.1). To check which: if differences are constant, it's linear; if ratios are constant, it's exponential. Example: 100, 150, 200, 250 has constant differences (+50) = linear. But 100, 110, 121, 133.1 has constant ratios (×1.1) = exponential! Let's contrast: Plan 1 involves adding the same amount each time—constant additive change. But Plan 2 involves multiplying by the same percent each time—constant multiplicative change. The exponential case has constant ratios (like ×1.05 each month), while a linear case would have constant differences (like +50 each month). This is exponential growth for Plan 2 and linear for Plan 1! Choice C correctly identifies this as Plan 1 linear (constant additive change) and Plan 2 exponential growth (constant percent change) because adding a fixed amount is linear, while percent increase is exponential. Choice D says both are exponential because they increase each month, but that's confusing the outcome (increase) with the mechanism: constant addition = linear, constant multiplication = exponential! Exponential vs linear quick-check: calculate both differences AND ratios. If differences are constant (like +5, +5, +5), it's linear. If ratios are constant (like ×1.1, ×1.1, ×1.1), it's exponential. Can't be both! This two-part check prevents confusion between the types. Real-world clue: exponential growth/decay contexts involve processes where the amount of change depends on how much you currently have: 'Population grows by 5% per year' means a population of 1000 gains 50, but a population of 10,000 gains 500—the change is bigger when the base is bigger. That's exponential! Linear is when you add the same amount regardless of current size.