1.13 Function model selection and assumption articulation
Identify the most suitable type of function to model the data sets.
| Data Sets | Type of Function | |
|---|---|---|
| 1 | (-3, 5), (-2, 7), (-1, 9), (0, 11), (1, 13) | |
| 2 | (0, 2), (1, 6), (2, 18), (3, 54) | |
| 3 | (1, 1), (2, 4), (3, 9), (4, 16), (5, 25) | |
| 4 | (1, 3), (2, 5), (4, 9), (6, 13) |
Identify the most suitable type of function to model the situation described in each scenario.
A ball is thrown into the air and its height is measured over time.
The number of bacteria in a petri dish doubles every hour.
The cost of producing a certain item decreases as the number of items produced increases.
The relationship between the temperature outside and the number of people at a park.
Given the following piecewise-defined function, state the interval for each function:
f(x) = \begin{cases} x^2 & \text{for } x < 0 \\ 3x + 5 & \text{for } 0 \leq x < 3 \\ 4 & \text{for } x \geq 3 \end{cases}For each function:
Find the domain and range of the function.
Identify any restrictions on the domain and range.
For each function, determine whether the function represents an odd-degree or even-degree polynomial.
Explain why a quadratic function would be more suitable to model the trajectory of a thrown ball than a linear function.
A scientist is studying the relationship between the concentration of a substance and the rate of a chemical reaction. The rate of the reaction increases as the concentration of the substance increases, but with diminishing returns. Explain why a rational function would be a suitable choice to model this scenario.
The temperature of a substance is modeled by the following piecewise-defined function:
T\left(t\right) = \begin{cases} -t, & 0 \leq t \lt 3 \\ 2t-6, & 3 \leq t \leq 5 \\ 4, & t \gt 5 \end{cases}where T is the temperature in degrees Celsius and t is the time in hours.
Determine the temperature of the substance after 2 hours.
At what time does the substance reach a temperature of 4 degrees Celsius?
The graph of a function modeling the number of hours of daylight h(x) throughout the year in a city is shown. Determine the appropriate domain and range restrictions for this model.
A company's profit, P(x), in thousands of dollars, can be modeled by the cubic function P(x) = x^3 - 6x^2 + 9x, where x represents the number of products sold in thousands. Determine the appropriate domain and range restrictions for this model.
The table shows the number of people who attended a concert each day for a week.
| Day | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Attendance | 100 | 200 | 300 | 350 | 300 | 200 | 150 |
A linear model was proposed to represent the attendance: A(x) = 50x + 50
Modify the model to better fit the given data set. Explain your reasoning.
A quadratic model was proposed to represent the profit of a company based on the number of products sold: P(x) = -0.01x^2 + 2x - 50
It was observed that the profit increases at a faster rate when more than 100 products are sold. Modify the model to better fit this observation. Explain your reasoning.
For every scenario, determine the minimum degree of the polynomial function that could model the given behavior. Assume all polynomial functions have real coefficients.
A roller coaster track has 3 peaks and 2 valleys throughout its course.
A graph showing the number of daylight hours as a function of the day of the year has exactly two maximum points and two minimum points.
A graph showing the revenue of a company over the years has 5increases and 4 decreases.
A diagram of the changes in a population of a species over time, showing three periods of growth and two periods of decline.
The function g(x) = \sqrt{x} + 5 models the growth of a plant over time. If time is measured in weeks and the plant cannot grow beyond 30 \text{ cm}, explain why a range restriction would be suitable and state what that restriction might be.
The function p(x) = \dfrac{1}{x} models the speed of a car based on the amount of fuel remaining. If the speed of the car cannot exceed 120 \text{ km/hr}, state the range restriction that should be applied to the function.
A plant biologist is studying the rate of growth of a particular type of plant. She finds that over a period of 30 days, the growth of the plant can be modelled by the function h(t)=0.3t^2+0.5t+1, where h(t) is the height of the plant in inches and t is the time in days.
Interpret the parameters of the function in the context of this problem. What does each term in the function represent in terms of the plant's growth?
A city's population, P, in thousands, can be modelled by the function P(t) = 500(1.02)^t, where t is the number of years since the year 2000.
Interpret the parameters of the function in the context of this problem. What does each term in the function represent in terms of the city's population growth?
A company's profit, P, in thousands of dollars, can be modelled by the function \\P(x) = 50x - 0.5x^2, where x is the number of units produced and sold.
Using this model, determine the number of units the company needs to produce and sell in order to maximize profit.
The graph shows the height of a projectile as a function of time. A cubic model is proposed to represent the height as a function of time.
Evaluate the fit of the cubic model to the data set. Explain your reasoning.
The population of a city is modeled by the following exponential function, where P is the population and t is the number of years since 2000:
P(t) = 5000 \cdot 1.03^tThe actual population of the city in 2010 was 7500. Evaluate the fit of the exponential model to the actual data. Explain your reasoning.
A logistic model is proposed to represent the growth of bacteria in a petri dish. The following data is provided:
| Time (hours) | 0 | 2 | 4 | 6 | 8 | 10 |
|---|---|---|---|---|---|---|
| Bacteria Count | 50 | 100 | 200 | 350 | 450 | 490 |
Evaluate the fit of the logistic model to the data set. Explain your reasoning.
The table shows the number of people who attended a local concert over a period of 9 consecutive years. The numbers represent thousands of people.
| Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| Number of Attendees (in thousands) | 2 | 5 | 11 | 20 | 32 | 47 | 65 | 86 | 110 |
Based on the data presented, which type of function do you think would best model the relationship between the year and the number of attendees? Justify your answer.
Consider the following scenario: A web development company charges a fixed amount for the initial website setup and then a monthly maintenance fee. Over time, the total cost of having a website developed and maintained by this company can be modelled by the function f(x) = 100x + 500, where x is the number of months and f(x) is the total cost in dollars.
Identify the type of function that models the scenario.
Explain why it is the most suitable the type of function that models the scenario.
Describe any domain and range restrictions that should be applied in the context.
A theme park has different ticket prices for different ages. Children under 12 are charged \$20, people aged 12 – 59 are charged \$50, and seniors aged 60 and above are charged \$30.
Write a piecewise-defined function to model this scenario. Be sure to clearly define the domain for each piece of the function.
A population of bacteria in a petri dish grows according to the modelP(t) = 500(1.2)^t, where P(t) is the population size and t is the time in hours.
Explain the behavior of this model in terms of its degree and type. Given the context, suggest appropriate range restrictions and explain your reasoning.