AP Precalculus Quiz: Competing Function Model Validation

A town tracks new streaming-service subscribers each month for one year. Model 1: Exponential, $S(t)=120(1.18)^t$, uses base $1.18$ to represent steady 18% monthly growth. Model 2: Logarithmic, $S(t)=80+140\log(0.6t+1)$, uses coefficient $140$ to reflect early buzz that slows as the market fills. By month 10, the actual totals rise only a few subscribers per month, not dozens. Based on the models described in the passage, which model best predicts long-term behavior in the context described?

A school tracks users of a new homework app each week. Model 1: Exponential, $U(t)=40(1.30)^t$, uses base $1.30$ for rapid word-of-mouth growth. Model 2: Logarithmic, $U(t)=35+120$, uses coefficient to model fast early adoption that slows as most students join. After week 8, nearly everyone who wants the app already has it, and weekly increases shrink. Using the information from the passage, why might the logarithmic model be more appropriate than the exponential model in this scenario?

A lab measures product formed after mixing chemicals: output rises quickly, then each extra minute adds less product. Model 1: Exponential, $y=5(1.25)^t$ (growth factor $1.25$). Model 2: Logarithmic, $y=18+7\log(2t+1)$ (coefficient ). The exponential model fits early rapid change, while the logarithmic model reflects a slowing rate as the reaction nears completion. Based on the models described in the passage, why might the logarithmic model be more appropriate than the exponential model in this scenario?

A town studies long-term population, but planners expect crowding to slow growth over time. Model 1: Exponential, $y=20000(1.04)^x$, where 1.04 is the annual growth factor. Model 2: Logarithmic, $y=20000+7000\log(x+1)$, where 7000 controls the added increase. Early measurements rise quickly and the exponential model matches the first few years well. After year 12, the yearly increase shrinks as housing becomes scarce. The exponential model keeps accelerating and predicts very large populations by year 50. The logarithmic model still rises but flattens, aligning with the crowding constraint. Based on the models described in the passage, which model best predicts long-term behavior in the context described?

A town’s population rises after a new highway opens, but land and water limits slow later growth. Model 1: Exponential, $y=12000(1.05)^t$ (growth factor $1.05$). Model 2: Logarithmic, $y=10000+3500\log(0.5t+1)$ (coefficient ). Early years show near-constant percent increases, but later years add fewer people each year. Using the information from the passage, which model best predicts long-term behavior in the context described?

A company monitors adoption of a new device, with many early buyers and fewer later as interest fades. Model 1: Exponential, $y=600(1.5)^x$, where 1.5 is the monthly growth factor. Model 2: Logarithmic, $y=600+900\log(x+1)$, where 900 scales the increase. During the first two months, sales jump sharply and resemble exponential growth. By month 9, sales still rise but with noticeably smaller month-to-month gains. The exponential model predicts extremely high adoption by month 18, exceeding the company’s estimated market size. The logarithmic model increases while slowing, matching the later trend. Using the information from the passage, why might the logarithmic model be more appropriate than the exponential model in this scenario?

A savings plan shows strong early gains, but a policy caps yearly interest so increases taper off. Model 1: Exponential, $y=5000(1.06)^t$ (base $1.06$). Model 2: Logarithmic, $y=5000+2200\log(0.9t+1)$ (coefficient ). Both models can match the first few years, yet later deposits produce smaller added returns. Using the information from the passage, which model best predicts long-term behavior in the context described?

A bank account balance is recorded yearly. Model 1: Exponential, $V(t)=2000(1.09)^t$, uses base $1.09$ for 9% annual compounding. Model 2: Logarithmic, $V(t)=2000+1500$, uses coefficient to represent gains that slow as opportunities diminish. In later years, the actual balance grows less each year than earlier, despite continued saving. Based on the models described in the passage, what are the implications of choosing an exponential model over a logarithmic model for long-term predictions?

A savings account balance is recorded yearly, showing strong early gains but smaller increases later as fees reduce effective returns. Model 1: Exponential, $y=2000(1.08)^x$, where 1.08 is the annual growth factor. Model 2: Logarithmic, $y=2000+900\log(x+1)$, where 900 sets the gain scale. After year 6, the balance still rises, but each added year contributes less than before. The exponential model matches years 1–3 closely, yet it projects unrealistically large balances by year 20. The logarithmic model reflects diminishing gains while still increasing over time. Using the information from the passage, why might the logarithmic model be more appropriate than the exponential model in this scenario?

A city’s population is estimated each year after a new factory opens. Model 1: Exponential, $P(t)=50{,}000(1.04)^t$, uses base $1.04$ for steady 4% growth. Model 2: Logarithmic, $P($, uses coefficient to reflect growth that slows as housing becomes scarce. After 15 years, reports show crowded housing and smaller annual increases than before. Based on the models described in the passage, which model best predicts long-term behavior in the context described?

A city models population over decades, but officials note limited housing and water supply. Model 1: Exponential, $y=50000(1.03)^x$, where 1.03 is the decade growth factor. Model 2: Logarithmic, $y=50000+18000\log(x+1)$, where 18000 controls the increase size. Early census counts rise quickly and resemble the exponential curve for two decades. After that, growth slows as new building permits become scarce. The exponential model keeps accelerating and predicts extreme populations by year 80. The logarithmic model still increases but at a decreasing rate, consistent with resource limits. Based on the models described in the passage, what are the implications of choosing an exponential model over a logarithmic model for long-term predictions?

A country estimates population after a drought reduces available farmland. Model 1: Exponential, $P(t)=3.2\text{ million}(1.03)^t$, uses base $1.03$ for 3% annual growth. Model 2: Logarithmic, $P($, uses coefficient million for slowing growth as resources tighten. Later reports show smaller annual increases and pressure on food supply. Using the information from the passage, why might the logarithmic model be more appropriate than the exponential model in this scenario?

A factory tests a catalyst: output rises sharply at low doses, then extra catalyst adds less improvement. Model 1: Exponential, $y=10(1.4)^d$ (base $1.4$). Model 2: Logarithmic, $y=25+12\log(1.2d+1)$ (coefficient ). Exponential growth seems plausible for strong early effects, while logarithmic growth reflects saturation of available reaction sites. Based on the models described in the passage, why might the logarithmic model be more appropriate than the exponential model in this scenario?

A controlled bacterial culture grows quickly in the first hours, but nutrients run low and the hourly increase shrinks. Model 1: Exponential, $y=800(1.3)^t$ (growth factor $1.3$). Model 2: Logarithmic, $y=900+500\log(0.7t+1)$ (coefficient ). The exponential model suits unchecked reproduction, while the logarithmic model suggests growth slows under limits. Based on the models described in the passage, what are the implications of choosing an exponential model over a logarithmic model for long-term predictions?

An investment account grows quickly at first, but the bank adds smaller bonuses after the balance gets large. Model 1: Exponential, $y=1000(1.08)^t$ (base $1.08$). Model 2: Logarithmic, $y=1000+900\log(0.5t+1)$ (coefficient ). The first model matches constant-percent interest, while the second matches diminishing returns from capped bonuses. Based on the models described in the passage, which model best predicts long-term behavior in the context described?

A new phone game gets downloads that double early, but later weeks add fewer downloads as interest fades. Model 1: Exponential, $y=200(1.6)^w$ (base $1.6$). Model 2: Logarithmic, $y=300+250\log(0.8w+1)$ (coefficient ). Early points fit both, yet the later data show smaller weekly gains. Based on the models described in the passage, in what situation would the exponential model provide a better fit than the logarithmic model?

A streaming service tracks total subscribers, seeing a burst of sign-ups that gradually slows as the market fills. Model 1: Exponential, $y=8000(1.25)^x$, where 1.25 is the monthly growth factor. Model 2: Logarithmic, $y=8000+12000\log(x+1)$, where 12000 sets the growth scale. The first four months show increases close to the exponential prediction. By month 10, the monthly increase is much smaller than earlier, hinting at saturation. The exponential model continues predicting rapidly rising totals, exceeding realistic market size. The logarithmic model still grows but with smaller increments, aligning with the later data. Using the information from the passage, why might the logarithmic model be more appropriate than the exponential model in this scenario?

An investor follows a new trading strategy and records account value monthly. Model 1: Exponential, $A(t)=5000(1.12)^t$, uses base $1.12$ for 12% monthly growth. Model 2: Logarithmic, $A(t)=5000+3200$, uses coefficient to represent diminishing gains as easy opportunities disappear. After several months, the added value per month becomes smaller rather than larger. Based on the models described in the passage, what are the implications of choosing an exponential model over a logarithmic model for long-term predictions?

A wildlife reserve counts a species each year, but rangers report food shortages limiting future growth. Model 1: Exponential, $y=300(1.5)^x$, where 1.5 is the yearly growth factor. Model 2: Logarithmic, $y=300+220\log(x+1)$, where 220 scales the increase. In years 1–2, the population rises quickly and resembles the exponential pattern. By year 6, growth slows noticeably, and the counts increase by smaller amounts each year. The exponential model predicts explosive growth by year 12, conflicting with limited habitat. The logarithmic model rises but flattens, consistent with environmental constraints. Using the information from the passage, in what situation would the exponential model provide a better fit than the logarithmic model?

A streaming platform sees new subscribers surge after a celebrity endorsement, then growth slows as most interested people have joined. Model 1: Exponential, $y=300(1.5)^x$ (base $1.5$). Model 2: Logarithmic, $y=500+400\log(0.4x+1)$ (coefficient ). Both can match early data, but later weekly increases shrink noticeably. Using the information from the passage, why might the logarithmic model be more appropriate than the exponential model in this scenario?