Function Model Selection and Assumption Articulation
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AP Precalculus › Function Model Selection and Assumption Articulation
Data from an experiment shows a relationship where the output variable increases to a single maximum value and then decreases, appearing to be symmetric. Both a quadratic and a quartic ($$4^{th}$$ degree) polynomial model fit the data well. In the absence of a theoretical reason to prefer one over the other, why might a researcher choose the quadratic model?
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.
The quadratic model is chosen only if its leading coefficient is positive, ensuring the function opens upwards to match the symmetric data.
The quadratic model is often preferred because it is a simpler model that still captures the essential features of the data (one maximum, symmetry).
The choice is arbitrary because both models fit the data well, and their predictions will be effectively identical for all possible input values.
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.
The height $$h$$, in meters, of a ball thrown upwards from a building is modeled by the function $$h(t) = -4.9t^2 + 20t + 50$$, 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?
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.
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.
The domain does not need any restriction because a quadratic function is defined for all real numbers and provides a complete path.
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 $$t=0$$.
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?
An exponential function, because the initial increase in yield is often rapid, suggesting a multiplicative growth factor.
A cubic function, because it can model both increasing and decreasing behavior in a single smooth curve with multiple inflection points.
A linear function, as long as the farmer only uses amounts of fertilizer on the increasing portion of the effectiveness curve.
A quadratic function, because its parabolic shape can effectively model a relationship with a single maximum point and symmetric behavior.
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.
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?
The sunflower's growth will eventually slow down and stop, reaching a maximum height that is implicitly determined by the linear model.
The initial height of the sunflower was 0.5 centimeters at the moment it sprouted, which corresponds to the rate of change.
The sunflower grows at a constant rate of 0.5 centimeters per day throughout the entire period being modeled by the function.
The sunflower's height increases by a larger amount each day, which is characteristic of accelerated growth patterns in young plants.
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.
An open-top box is to be made from a square piece of cardboard measuring 24 inches on each side by cutting equal squares of side length $$x$$ from each of the four corners and folding up the sides. Which function type best models the volume, $$V$$, of the box as a function of $$x$$?
A rational function, because the process involves dividing the original cardboard into smaller sections to form the box.
A linear function, because the side length $$x$$ is a linear measure and directly relates to the dimensions of the final box.
A quadratic function, because the base of the box is a square, and the area of a square is a quadratic relationship.
A cubic function, because the volume is the product of three linear dimensions (length, width, and height) that are all functions of $$x$$.
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.
A farmer wants to build a rectangular fence for a garden using 100 feet of fencing. Which function type best models the area, $$A$$, of the garden as a function of its length, $$l$$?
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.
A piecewise-defined function, because the length and width must be positive, which introduces constraints on the possible dimensions of the garden.
A cubic function, because the problem involves maximizing a quantity within a constraint, which often leads to cubic models in optimization.
A linear function, because the perimeter is a linear quantity and directly determines the 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.
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?
A quadratic function, because production costs often involve economies of scale, leading to a non-constant rate of change.
A linear function, because a nearly constant cost for each additional unit implies a nearly constant rate of change (slope).
An exponential function, because if the cost of materials increases over time, the total cost could grow exponentially with production.
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.
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?
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.
The model will eventually predict a population of zero, which is unlikely for a thriving bacterial culture unless specific conditions are met.
The model assumes the growth rate is constant, which is a characteristic of linear models, not cubic models for bacterial populations.
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.
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?
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.
A linear function, because the cost increases for data usage beyond the initial amount included, indicating a generally increasing trend.
A polynomial function of degree 3, because there is an initial flat fee followed by a variable charge, requiring a more complex model.
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.
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.
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?
Rational $P(t)=\frac{at+b}{ct+d}$ with horizontal asymptote $40$
Quartic polynomial with three turning points and unbounded ends
Linear polynomial with constant net increase per year
Quadratic polynomial $P(t)=at^2+bt+c$ opening upward forever
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.