- A polynomial function C(t) is used to model the concentration of a medication in a patient's bloodstream, in mg/L, at time t hours after administration. Which of the following assumptions about the domain and range of C(t) is most critical for the model to be physically realistic?
Answer: The domain must be restricted to t≥0 and the range must be restricted to C(t)≥0, as time and concentration cannot be negative values.
For the model to be realistic, the input variable, time t, cannot be negative. The output variable, concentration C(t), also cannot be negative. Therefore, restricting the domain and range to non-negative values is a critical assumption. - A company models profit P(q)=R(q)−C(q) with R(q) linear in q and C(q) convex. Why is a polynomial model more appropriate than a rational model in this context?
Answer: Profit should be smooth with no vertical asymptotes for q≥0
This question tests AP Precalculus skills in selecting appropriate function models and articulating necessary assumptions for polynomial and rational functions. Polynomial functions are smooth and continuous, suitable for modeling scenarios with constant rates of change, while rational functions handle asymptotic behavior and discontinuities. In this scenario, profit is the difference between linear revenue and convex cost functions, resulting in a smooth function without discontinuities. Choice A is correct because profit functions in business contexts should be continuous and defined for all non-negative quantities, which polynomial models guarantee. Choice B is incorrect because requiring a nonzero horizontal asymptote would mean profit approaches a fixed value as production increases indefinitely, which contradicts typical business scenarios where costs eventually exceed revenue. Encourage students to consider practical constraints when selecting models and recognize that polynomial functions are ideal for smooth, continuous relationships. Practice analyzing why certain mathematical features (like asymptotes or discontinuities) may be inappropriate for specific contexts. - A botanist proposes a linear function H(d)=0.5d+2 to model the height of a sunflower, in centimeters, d days after it sprouted. What is a key assumption made in this linear model?
Answer: The sunflower grows at a constant rate of 0.5 centimeters per day throughout the entire period being modeled by the function.
A linear model of the form y=mx+b has a constant rate of change given by the slope m. In this model, the slope is 0.5, so the key assumption is that the sunflower's height increases at a constant rate of 0.5 cm per day. - A projectile's height data at t=0,1,2 s are 1.5,12.1,13.0 m, with no thrust after launch. Which function model best fits the given data?
Answer: Quadratic polynomial h(t)=at2+bt+c on 0≤t≤T
This question tests AP Precalculus skills in selecting appropriate function models and articulating necessary assumptions for polynomial and rational functions. Polynomial functions are smooth and continuous, suitable for modeling scenarios with constant rates of change, while rational functions handle asymptotic behavior and discontinuities. In this scenario, the projectile height data shows an initial rise (1.5 to 12.1 m) followed by deceleration (12.1 to 13.0 m), characteristic of quadratic motion under constant gravity. Choice A is correct because projectile motion without thrust follows h(t) = at² + bt + c where a < 0 represents gravitational acceleration, perfectly fitting the parabolic trajectory on the flight interval. Choice D is incorrect because gravity causes constant acceleration, not constant height change; the velocity changes linearly while height changes quadratically. Encourage students to recognize that free-fall motion always produces quadratic height functions. Practice connecting physical laws like constant acceleration to their corresponding polynomial models. - The effectiveness of a particular fertilizer is measured by crop yield. As the amount of fertilizer applied increases from zero, the yield increases up to a certain point, after which applying more fertilizer causes the yield to decrease. The data appears to be symmetric around the point of maximum yield. Which function type would be most appropriate for modeling the crop yield as a function of the amount of fertilizer applied?
Answer: A quadratic function, because its parabolic shape can effectively model a relationship with a single maximum point and symmetric behavior.
The description of the data—increasing to a single maximum and then decreasing symmetrically—is the classic behavior modeled by a downward-opening parabola, which is the graph of a quadratic function. This function type captures the single peak and symmetric decline effectively.
- A scientist has collected five distinct data points from an experiment. Assuming there is no experimental error, what is the minimum degree of a single polynomial function that is guaranteed to be able to pass through all five of these points?
Answer: Degree 4, because a polynomial of degree at most n can be found that passes through any n+1 distinct points.
For any set of n+1 points with distinct x-coordinates, there exists a unique polynomial of degree at most n that passes through all of them. For 5 points, we have n+1=5, which implies n=4. Therefore, a polynomial of degree 4 is the minimum degree guaranteed to fit the data. - A city's population P(t) (millions) is modeled with carrying capacity 3.2; data: P(0)=2.1, P(10)=2.6. What assumptions are necessary for this model to be valid?
Answer: Assume continuous time and a fixed carrying capacity over decades
This question tests AP Precalculus skills in selecting appropriate function models and articulating necessary assumptions for polynomial and rational functions. Polynomial functions are smooth and continuous, suitable for modeling scenarios with constant rates of change, while rational functions handle asymptotic behavior and discontinuities. In this scenario, the city population grows from 2.1 to 2.6 million over 10 years with a carrying capacity of 3.2 million, indicating logistic growth. Choice A is correct because logistic models assume continuous time (not discrete jumps) and a fixed carrying capacity that limits growth over the modeling period. Choice D is incorrect because constant net change (linear growth) would ignore the carrying capacity constraint and predict the population exceeding 3.2 million eventually. Encourage students to recognize that carrying capacity models require assumptions about resource limitations remaining constant. Practice connecting demographic constraints to mathematical model assumptions and their implications.
- A candle is initially 12 inches tall. It burns at a constant rate of 0.5 inches per hour. Which type of function would be most appropriate to model the height of the candle as a function of the time it has been burning?
Answer: A linear function, because the height decreases by a constant amount for each unit of time, which corresponds to a constant rate of change.
The phrase "burns at a constant rate" directly translates to a constant rate of change, which is the slope of a linear function. The height decreases by 0.5 inches for every 1 hour that passes, which defines a line with a slope of -0.5.
- A lake's fish population P(t) (thousands) satisfies P(0)=18, P(4)=26, and levels near 40 due to resources. Which function model best fits the given data?
Answer: Rational P(t)=ct+dat+b with horizontal asymptote 40
This question tests AP Precalculus skills in selecting appropriate function models and articulating necessary assumptions for polynomial and rational functions. Polynomial functions are smooth and continuous, suitable for modeling scenarios with constant rates of change, while rational functions handle asymptotic behavior and discontinuities. In this scenario, the fish population grows from 18 to 26 thousand over 4 years and levels near 40 thousand due to resource limitations, indicating logistic growth behavior. Choice A is correct because a rational function with horizontal asymptote at 40 can model the population approaching but never exceeding the carrying capacity. Choice B is incorrect because a quadratic opening upward would predict unbounded growth, contradicting the resource-limited leveling behavior. Encourage students to recognize carrying capacity as a key indicator for rational models with horizontal asymptotes. Practice connecting biological constraints like limited resources to mathematical features like asymptotic behavior. - A biologist uses a cubic polynomial function to model the population of a certain bacteria culture over a 12-hour period. The model is a good fit for the experimental data collected during these 12 hours. Which of the following is a key limitation of using this polynomial model to predict the population for times far beyond the 12-hour period?
Answer: The model's end behavior approaches positive or negative infinity, which is not a realistic long-term behavior for a population in a finite environment.
A non-constant polynomial function has end behavior that approaches either positive or negative infinity. A real-world population is constrained by its environment (e.g., food, space) and cannot grow infinitely. This makes the polynomial model unsuitable for long-term predictions outside its initial observation window.
- A projectile's height h(t) (ft) follows h(0)=6, h(0.5)=38, h(1)=54, h(2)=6 under constant gravity. How does the behavior of this function at infinity affect its suitability?
Answer: Quadratic is suitable since h(t)→−∞ as t→∞
This question tests AP Precalculus skills in selecting appropriate function models and articulating necessary assumptions for polynomial and rational functions. Polynomial functions are smooth and continuous, suitable for modeling scenarios with constant rates of change, while rational functions handle asymptotic behavior and discontinuities. In this scenario, the given data suggests a pattern best modeled by a quadratic polynomial, as indicated by the symmetric parabolic path where h(0) = h(2) = 6 feet with a maximum in between. Choice A is correct because it recognizes that under constant gravity, projectile height follows a quadratic function that approaches negative infinity as time increases (the object continues falling below ground level in the mathematical model). Choice B is incorrect because projectile motion doesn't have horizontal asymptotes - the height continues decreasing without bound. Encourage students to match model features with data trends and verify assumptions necessary for model validity. Practice understanding that physical constraints (like the ground) are separate from the mathematical model's behavior. - A set of data relates an independent variable x to a dependent variable y. When analyzing the differences in y-values for uniform increases in x-values, it is found that the first differences are not constant, but the second differences are constant and non-zero. Which type of function is the most appropriate to model this data?
Answer: A quadratic function, because constant second differences are a defining characteristic of quadratic relationships.
A key property of polynomial functions is that a polynomial of degree n has constant n-th differences for uniform changes in the input variable. Since the second differences are constant, a quadratic function (degree 2) is the most appropriate model. - A taxi company charges a flat fee of 3.00foranytrip,plus2.00 per mile for the first 10 miles. For any miles traveled beyond 10 miles, the company charges $1.50 per mile. Which function type is most appropriate for modeling the total cost of a taxi ride as a function of the number of miles traveled?
Answer: A piecewise-defined function, because the per-mile rate, and thus the rule for calculating the total cost, changes at the 10-mile mark.
The cost calculation is different for trips up to 10 miles compared to trips longer than 10 miles. This change in the function's rule based on the input value (miles) necessitates a piecewise-defined function to accurately model the total cost.
- A projectile's height h(t) (m) satisfies h(0)=1.5, h(1)=22, h(2)=32, h(4)=1.5 with negligible air resistance. Why is a polynomial model more appropriate than a rational model in this context?
Answer: A polynomial model matches constant-acceleration curvature
This question tests AP Precalculus skills in selecting appropriate function models and articulating necessary assumptions for polynomial and rational functions. Polynomial functions are smooth and continuous, suitable for modeling scenarios with constant rates of change, while rational functions handle asymptotic behavior and discontinuities. In this scenario, the given data suggests a pattern best modeled by a polynomial function, as indicated by the symmetric parabolic path of a projectile under constant gravity with negligible air resistance. Choice B is correct because it aligns with the physics principle that projectile motion under constant acceleration follows a quadratic path, matching the constant-acceleration curvature. Choice C is incorrect because boundedness is not required for projectile motion - the height simply returns to ground level at a finite time. Encourage students to match model features with data trends and verify assumptions necessary for model validity. Practice connecting physical principles to mathematical models, recognizing that constant acceleration produces quadratic position functions.
- A person's journey is described as follows: they walk away from home at a constant speed for 10 minutes, then stand still for 5 minutes, and finally walk back home at the same constant speed. Which function type would best model the person's distance from home as a function of time?
Answer: A piecewise-defined function, because the rate of change of distance from home is different during the three distinct phases of the journey.
The rate of change of distance (speed relative to home) is positive and constant for the first 10 minutes, zero for the next 5 minutes, and negative and constant for the final phase. Since the function's definition changes at different time intervals, a piecewise-defined function is necessary.
- A car starts from rest and accelerates such that its speed increases at a constant rate. Which function type best models the total distance the car has traveled as a function of time?
Answer: A quadratic function, because a constant acceleration (which implies a linear velocity) results in a distance that changes with the square of time.
If speed increases at a constant rate, this means acceleration is constant. This implies the velocity function is linear (v(t)=at). The average rates of change of a quadratic function are linear. Therefore, a quadratic function is the appropriate model for the distance traveled over time. - A mobile phone plan charges a flat fee of 20 per month, which includes 5 gigabytes (GB) of data. For each gigabyte of data used beyond 5 GB, an additional fee of 10 is charged. Which function type best models the total monthly cost, C, as a function of the data used, d, in gigabytes?
Answer: A piecewise-defined function, because the rule for calculating the cost is different for data usage up to 5 GB versus data usage beyond 5 GB.
The cost is constant (C(d)=20) for 0≤d≤5 and then increases linearly (C(d)=20+10(d−5)) for d>5. Because the rule that defines the function changes at d=5, a piecewise-defined function is the most appropriate model. - A reversible reaction's concentration C(t) (mol/L) satisfies C(0)=0.00, C(2)=0.48, C(6)=0.73, and approaches 0.80 at equilibrium. What assumptions are necessary for this model to be valid?
Answer: Continuity and a horizontal asymptote as t→∞
This question tests AP Precalculus skills in selecting appropriate function models and articulating necessary assumptions for polynomial and rational functions. Polynomial functions are smooth and continuous, suitable for modeling scenarios with constant rates of change, while rational functions handle asymptotic behavior and discontinuities. In this scenario, the given data suggests a pattern best modeled by a rational function approaching equilibrium, as indicated by the concentration starting at 0 and approaching 0.80 mol/L. Choice A is correct because it identifies the key assumptions of continuity (no sudden jumps in concentration) and a horizontal asymptote at the equilibrium value of 0.80. Choice C is incorrect because there's no reason for a vertical asymptote - the reaction continues smoothly without any discontinuity at t=6. Encourage students to match model features with data trends and verify assumptions necessary for model validity. Practice recognizing that chemical equilibrium processes typically approach limiting values smoothly, making rational functions with horizontal asymptotes ideal models. - The height h, in meters, of a ball thrown upwards from a building is modeled by the function h(t)=−4.9t2+20t+50, where t is the time in seconds after the ball is thrown. Which of the following describes a necessary restriction on the domain of the function for this model to be physically realistic?
Answer: The domain must be restricted to t≥0 and end when the ball hits the ground, as time cannot be negative and the model is invalid after impact.
The context of the problem begins at t=0. Negative time is not meaningful. The model also ceases to be valid once the ball hits the ground (when h(t)=0 for some t>0). Therefore, the domain must be restricted to a closed interval starting at t=0. - A data set shows the cost of producing a certain number of items. An analysis of the data indicates that the cost to produce each additional item is approximately the same. Which function type is the most appropriate to model the total production cost as a function of the number of items produced?
Answer: A linear function, because a nearly constant cost for each additional unit implies a nearly constant rate of change (slope).
The phrase "cost to produce each additional item is approximately the same" is a description of the rate of change of the total cost function. A constant rate of change is the defining characteristic of a linear function. Therefore, a linear model is the most appropriate choice.
