AP Precalculus Quiz: Function Model Selection And Assumption Articulation

What this quiz covers

This quiz focuses on Function Model Selection And Assumption Articulation, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Precalculus.

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.

Question 1

1. The quartic model is always a better choice because its higher degree allows it to capture more complex variations that might exist in the data.
2. The quadratic model is often preferred because it is a simpler model that still captures the essential features of the data (one maximum, symmetry). (correct answer)
3. The choice is arbitrary because both models fit the data well, and their predictions will be effectively identical for all possible input values.
4. The quadratic model is chosen only if its leading coefficient is positive, ensuring the function opens upwards to match the symmetric data.

Explanation: The principle of parsimony suggests that when multiple models fit data well, the simplest model is generally preferred. A quadratic function is simpler (degree 2) than a quartic function (degree 4) and adequately describes the key features of the data: a single maximum and symmetric behavior.

Question 2

1. The domain must be restricted to $t \ge 0$ and end when the ball hits the ground, as time cannot be negative and the model is invalid after impact. (correct answer)
2. The domain must be restricted to values of $t$ for which $h(t) > 50$, because the ball is thrown upwards from an initial height of 50 meters.
3. The domain does not need any restriction because a quadratic function is defined for all real numbers and provides a complete path.
4. The domain must be restricted to exclude the time when the ball is at its maximum height, because the velocity is zero at that point.

Explanation: 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 $t=0$.

Question 3

1. A linear function, as long as the farmer only uses amounts of fertilizer on the increasing portion of the effectiveness curve.
2. A cubic function, because it can model both increasing and decreasing behavior in a single smooth curve with multiple inflection points.
3. A quadratic function, because its parabolic shape can effectively model a relationship with a single maximum point and symmetric behavior. (correct answer)
4. An exponential function, because the initial increase in yield is often rapid, suggesting a multiplicative growth factor.

Explanation: 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.

Question 4

1. The sunflower's height increases by a larger amount each day, which is characteristic of accelerated growth patterns in young plants.
2. The sunflower's growth will eventually slow down and stop, reaching a maximum height that is implicitly determined by the linear model.
3. The sunflower grows at a constant rate of 0.5 centimeters per day throughout the entire period being modeled by the function. (correct answer)
4. The initial height of the sunflower was 0.5 centimeters at the moment it sprouted, which corresponds to the rate of change.

Explanation: 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.

Question 5

1. A linear function, because the side length $x$ is a linear measure and directly relates to the dimensions of the final box.
2. A quadratic function, because the base of the box is a square, and the area of a square is a quadratic relationship.
3. A cubic function, because the volume is the product of three linear dimensions (length, width, and height) that are all functions of $x$. (correct answer)
4. A rational function, because the process involves dividing the original cardboard into smaller sections to form the box.

Explanation: The height of the box is $x$. The length and width of the base are both $24 - 2x$. The volume is $V(x) = (24 - 2x)(24 - 2x)(x)$, which is a cubic polynomial function. Geometric contexts involving volume often lead to cubic models.

Question 6

1. A linear function, because the perimeter is a linear quantity and directly determines the dimensions of the garden.
2. A quadratic function, because the area is the product of two linear dimensions which are dependent on each other, resulting in a single maximum area. (correct answer)
3. A cubic function, because the problem involves maximizing a quantity within a constraint, which often leads to cubic models in optimization.
4. A piecewise-defined function, because the length and width must be positive, which introduces constraints on the possible dimensions of the garden.

Explanation: Let the length be $l$. The perimeter is $2l + 2w = 100$, so the width is $w = 50 - l$. The area is $A(l) = l \times w = l(50 - l) = 50l - l^2$. This is a quadratic function. Its graph is a parabola, which correctly models the area increasing to a maximum and then decreasing.

Question 7

1. A quadratic function, because production costs often involve economies of scale, leading to a non-constant rate of change.
2. A linear function, because a nearly constant cost for each additional unit implies a nearly constant rate of change (slope). (correct answer)
3. An exponential function, because if the cost of materials increases over time, the total cost could grow exponentially with production.
4. A rational function, because the average cost per item changes as more items are produced, which is best represented by a ratio.

Explanation: 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.

Question 8

1. The model assumes the growth rate is constant, which is a characteristic of linear models, not cubic models for bacterial populations.
2. The model will eventually predict a population of zero, which is unlikely for a thriving bacterial culture unless specific conditions are met.
3. 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. (correct answer)
4. The model is too simple because a polynomial of degree 3 can only have at most two local extrema within the observation period.

Explanation: 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.

Question 9

1. A linear function, because the cost increases for data usage beyond the initial amount included, indicating a generally increasing trend.
2. A quadratic function, because the rate of cost increase changes at the 5 GB threshold, which creates a curve in the graph of the cost.
3. 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. (correct answer)
4. A polynomial function of degree 3, because there is an initial flat fee followed by a variable charge, requiring a more complex model.

Explanation: The cost is constant ($C(d)=20$) for $0 \le d \le 5$ 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.

Question 10

1. Rational $P(t)=\frac{at+b}{ct+d}$ with horizontal asymptote $40$ (correct answer)
2. Quadratic polynomial $P(t)=at^2+bt+c$ opening upward forever
3. Quartic polynomial with three turning points and unbounded ends
4. Linear polynomial with constant net increase per year

Explanation: 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.

Question 11

1. Cubic polynomial $C(t)=at^3+bt^2+ct+d$
2. Rational $C(t)=\dfrac{at+b}{t+c}$ with horizontal asymptote (correct answer)
3. Quadratic polynomial $C(t)=at^2+bt+c$
4. Linear polynomial $C(t)=mt+b$

Explanation: 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, as indicated by the concentration increasing from 0.30 to 0.69 but expected to plateau below 0.75. Choice B is correct because it aligns with the data's growth pattern that slows over time and approaches a horizontal asymptote below 0.75, typical of chemical reactions approaching equilibrium. Choice C (quadratic) is incorrect because quadratic functions either grow without bound or have a maximum and then decrease, neither matching the plateauing behavior. Encourage students to match model features with data trends and verify assumptions necessary for model validity. Practice analyzing chemical concentration data that approaches equilibrium values asymptotically.

Question 12

1. Assume continuous time and a fixed carrying capacity over decades (correct answer)
2. Assume population is periodic with a 10-year cycle
3. Assume a vertical asymptote at $t=10$ due to migration
4. Assume constant net change so $P(t)$ is linear for all $t$

Explanation: 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.

Question 13

1. A rational model guarantees symmetry about the peak
2. A polynomial model matches constant-acceleration curvature (correct answer)
3. A rational model is required because $h(t)$ must be bounded
4. A polynomial model must have a vertical asymptote at landing

Explanation: 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.

Question 14

1. Continuity in $t$ and approach to a horizontal asymptote (correct answer)
2. A vertical asymptote at $t=12$ from overcrowding
3. Unbounded growth because births exceed deaths forever
4. Exact linear change each month over all time

Explanation: 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 carrying capacity, as indicated by the population growing from 500 to 1100 and stabilizing near 1200. Choice A is correct because it identifies the key assumptions of continuity (smooth population growth) and approach to a horizontal asymptote at the resource-limited carrying capacity of 1200. Choice B is incorrect because there's no reason for a vertical asymptote at t=12 - populations don't suddenly become infinite due to overcrowding. Encourage students to match model features with data trends and verify assumptions necessary for model validity. Practice recognizing that ecological models with resource limitations require functions with horizontal asymptotes.

Question 15

1. Quadratic is suitable since $h(t)\to-\infty$ as $t\to\infty$ (correct answer)
2. Rational is suitable since $h(t)$ must have a horizontal asymptote
3. Linear is suitable since $h(t)$ stays bounded for all $t$
4. Rational is suitable since $h(t)$ needs a vertical asymptote

Explanation: 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.

Question 16

1. Quadratic polynomial $h(t)=at^2+bt+c$ on $0\le t\le T$ (correct answer)
2. Rational $h(t)=\frac{at+b}{ct+d}$ with vertical asymptote at impact
3. Quartic polynomial to capture multiple bounces automatically
4. Linear polynomial because gravity produces constant height change

Explanation: 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.

Question 17

1. Continuity and a horizontal asymptote as $t\to\infty$ (correct answer)
2. Periodicity and repeating peaks every two minutes
3. A vertical asymptote at $t=6$ due to completion
4. Unbounded growth because reactants are unlimited

Explanation: 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.

Question 18

1. Profit should be smooth with no vertical asymptotes for $q\ge0$ (correct answer)
2. Profit must approach a nonzero horizontal asymptote as $q\to\infty$
3. Profit should be periodic because demand cycles weekly
4. Profit must be undefined at the break-even quantity

Explanation: 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.

Question 19

1. A quadratic function, because it can model a situation that increases and then decreases, capturing a single peak profit.
2. A linear function, because it can represent the overall trend of the profit change from the start to the end of the period.
3. A polynomial function of at least degree 3, because the profit has two turning points (a local maximum and a local minimum). (correct answer)
4. An exponential function, because profit models often exhibit growth that is proportional to the current profit.

Explanation: The description "increase... then decrease... and finally increase again" implies the existence of two turning points (extrema). A polynomial of degree $n$ can have at most $n-1$ turning points. To model two turning points, a polynomial of at least degree 3 is required.

Question 20

1. A linear function, because the relationship between the variables can often be reasonably approximated by a straight line.
2. A quadratic function, because constant second differences are a defining characteristic of quadratic relationships. (correct answer)
3. A cubic function, because the fact that the first differences are changing indicates a degree higher than linear is required.
4. An exponential function, because changing rates of change often suggest exponential growth or decay patterns.

Explanation: 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.