Exercises: Linear functions: interpolation

Look at the following graph, which represents the total cost of an order of custom T-shirts for an e-sports team as a function of the number of units purchased.

From the graph:

1. a) Identify the y-intercept ($n$) and explain its meaning in this context.

1. b) If the total cost for $10$ T-shirts is $50$ €, calculate the slope ($m$) of the line.

1. c) Write the analytical expression of the function $f(x)$.

An e-scooter rental company charges a fixed unlock fee and a constant rate for each minute of use. One user paid $4,50$ € for a $15$-minute trip, while another paid $7,50$ € for a $35$-minute trip.

Find the equation of the linear function relating the cost of the service ($y$) to the time of use in minutes ($x$).

The following table shows the temperature of an industrial oven as it heats up at a constant rate:

Use linear interpolation to estimate the temperature of the oven when $12$ minutes have elapsed since it was switched on.

A gardener observes that the growth of a species of bamboo is linear during its first few weeks. If at $10$ days it measured $45$ cm and at $20$ days it measured $75$ cm:

1. Determine the linear function that models the height of the bamboo as a function of time.

2. Perform a linear extrapolation to predict the height of the plant at $25$ days.

Find the linear function passing through the points $A(2, 7)$ and $B(5, 16)$. Once obtained, check whether the point $C(4, 13)$ belongs to this function using linear interpolation.

Two music platforms offer the following monthly plans:

• Plan A: $5$ € fixed fee plus $0,20$ € for each song downloaded.

• Plan B: No fixed fee, but $0,60$ € for each song downloaded.

Let $x$ be the number of songs downloaded in a month and $y$ the monthly cost in euros.

1. a) Write the cost function for each plan.

1. b) Graph both functions on the same Cartesian axes (it is enough to take values of $x$ between $0$ and $20$ songs) and find graphically the "break-even point" (where both plans cost the same). State its coordinates.

It is known that the relationship between the braking distance of a car ($d$) and its speed ($v$) is approximately linear for certain low-speed ranges. If at $20$ km/h the braking distance is $4$ metres and at $40$ km/h it is $12$ metres:

1. What would be the estimated braking distance for a speed of $30$ km/h?

2. If a car has left a braking mark of $10$ metres, approximately how fast was it travelling? (Use interpolation).

Given the following table of values relating the number of hours worked by a freelance programmer and their total fees:

Calculate, using the formula of the linear interpolation line, how much they would charge for a job of $7$ hours. Also state the hourly rate charged (gradient).

Check that the parabola $y = x^{2} - 2x + 3$ passes through the points $(0, 3)$, $(1, 2)$ and $(2, 3)$, and use it to estimate the value of $y$ at $x = 1,5$.

We want to interpolate the points $(0, 1)$, $(1, 4)$ and $(2, 9)$ using a parabola $y = ax^{2} + bx + c$.

1. a) Set up the system of three equations in unknowns $a$, $b$ and $c$.

1. b) Check that $a = 1$, $b = 2$ and $c = 1$ is a solution to the system and write the interpolating parabola.

In a study on the monthly cost of the youth travel pass in an autonomous community, two official data points are available: in 2018 ($t = 0$) it cost $20\text{ \euro}$ and in 2022 ($t = 4$) it cost $12\text{ \euro}$. Marcos wants to estimate the price it had in 2020 ($t = 2$) and the price it will have in 2026 ($t = 8$).

To do so, Marcos writes in his notebook:

«I calculate the slope: $m = \frac{4 - 0}{12 - 20} = \frac{4}{-8} = -0,5\text{ \euro/year}$. The line is $P(t) = -0,5t + 20$. Therefore, in 2020 it cost $P(2) = -0,5 \cdot 2 + 20 = 19\text{ \euro}$ and in 2026 it will cost $P(8) = -0,5 \cdot 8 + 20 = 16\text{ \euro}$. Both results are reliable interpolations because we followed the same mathematical formula».

1. a) Identify and explain the algebraic error Marcos made when setting up the slope. Write the equation of the correct interpolation line.
2. b) Using the correct line, calculate the price estimates for the years 2020 and 2026.
3. c) Correct Marcos's conceptual error: distinguish which of the two calculations is an interpolation and which is an extrapolation, and justify whether the prediction for 2026 has the same reliability as that for 2020 in this context.

A municipal water company in Andalusia records the average monthly consumption per household during a drought year. In May ($x = 5$, where $x$ represents the month number) the average consumption was $132\text{ m}^3$, while in August ($x = 8$) it rose to $168\text{ m}^3$ due to high temperatures. The technical team needs to estimate the average consumption for the month of June ($x = 6$).

1. a) Before carrying out any algebraic calculation, reasonably estimate between which two numerical values June's consumption should lie and argue whether it should be closer to $132\text{ m}^3$ or to $168\text{ m}^3$.
2. b) Find the equation of the linear interpolation line $C(x) = mx + n$ from the May and August data.
3. c) Analytically calculate with this line the predicted consumption for June ($x = 6$) and check whether the result matches the estimate you made in section a).
4. d) If that same line is used to predict the consumption for December ($x = 12$): is it an interpolation or an extrapolation? Explain why this result will hardly be reliable in reality.

A streaming content platform has analyzed the growth of its subscribers in a young city. In 2021 it had $12\,400$ users, and in 2023 the figure rose to $15\,200$.

If we assume that the growth follows a linear model, and we take $x$ as the time in years elapsed since 2021 (that is, $x=0$ for the year 2021, $x=2$ for 2023, and so on):

1. a) Find the linear interpolation function that expresses the number of subscribers as a function of time $x$.

1. b) Use interpolation to estimate the number of subscribers in mid-2022 (that is, for $x=1{,}5$).

1. c) Carry out an extrapolation to predict how many users there will be in 2026 and reflect on the reliability of this last figure.

The figure shows the two known data points (note that the vertical axis starts at $12\,000$).

A biology laboratory studies the breakdown of an organic compound. It has been observed that at a temperature of $15^\circ\text{C}$, the compound takes $40$ hours to decompose, whereas at $25^\circ\text{C}$ it takes only $28$ hours.

1. Determine the linear interpolation line relating the decomposition time ($y$) to the temperature ($x$).

2. What would be the estimated decomposition time if the temperature were $22^\circ\text{C}$?

3. If the laboratory needs the compound to degrade in $20$ hours, at what temperature should the thermostat be set according to this model?

The organisers of an urban music festival have observed the relationship between ticket price and the number of season tickets sold. If the price is $45$ €, $3\,200$ season tickets are sold. If the price rises to $60$ €, sales drop to $2\,450$ season tickets.

1. Find the equation of demand assuming it is a linear function.

2. Estimate how many season tickets would be sold if an "intermediate" price of $52$ € were set.

3. Above what price would demand be zero according to this extrapolation model?

The percentage of plastic packaging recycling in an autonomous region has evolved almost linearly over the last decade. In the year 2012, $18\,\%$ was recycled, and in the year 2020, $34\,\%$ was reached.

1. Calculate the mean annual rate of change of the recycling percentage between those years.

2. Use linear interpolation to estimate the percentage of recycling in the year 2015.

3. If the trend continues in the same way, in which year is the European Union target of $50\,\%$ expected to be reached?

An electric car draws energy from its battery in proportion to the distance traveled. When leaving the garage with $100\,\%$ charge, after traveling $120$ km, the battery shows $64\,\%$ charge remaining. The figure shows these two data points.

1. a) Write the linear function that gives the percentage of battery remaining as a function of the kilometers traveled.

1. b) If the driver decides to stop to recharge when $15\,\%$ of the battery is left, how many kilometers will they have traveled in total according to the linear model?

1. c) Using extrapolation, determine what percentage of battery the car would show after a $300$ km trip. Does the result make physical sense? Justify your answer.

In a sociological study on the use of digital devices, the following data were collected on the daily screen time (in minutes) spent by teenagers according to their age:

1. Determine the linear function that fits this data.

2. Estimate, by interpolation, the daily screen time for a $14$-year-old teenager.

3. If a $18$-year-old youth spends $250$ minutes a day, does this data point fit the linear model calculated above? Check it using extrapolation.

The total cost of producing custom t-shirts for a graduation includes a fixed set-up cost for the design and a variable cost per unit manufactured. We know that $50$ t-shirts cost $410$ € and that $120$ t-shirts cost $830$ €.

1. Calculate the cost of manufacturing $85$ t-shirts using the linear interpolation technique.

2. Identify from the resulting line the fixed cost (y-intercept) and the cost per additional t-shirt (gradient).

3. If the class's maximum budget is $600$ €, what is the maximum number of t-shirts they can order?

The snow depth at a ski resort is being measured during a day of continuous snowfall. At $8:00$ h the depth was $45$ cm and at $13:00$ h it had risen to $62$ cm.

1. Find the equation of the line relating snow depth ($y$) to the hours elapsed since $8:00$ h ($x$).

2. What snow depth was estimated at $11:30$ h?

3. If the snowfall began at $5:00$ h in the morning with the same intensity, how much snow was on the ground at that time? Use linear extrapolation.

The figure shows a parabola passing through the points $(1, 3)$, $(3, 7)$ and $(5, 5)$.

1. a) Set up the system of equations that $a$, $b$ and $c$ must satisfy so that $y = ax^{2} + bx + c$ passes through the three points, and solve it.

1. b) Use the parabola obtained to estimate the value of $y$ at $x = 4$ (the one marked with "?" in the figure).

1. c) Check with the graph that the estimate is consistent.

A concert promoter has tested three ticket prices with different results: at $10\ \text{ €}$ it sold $400$ tickets; at $20\ \text{ €}$, $700$; and at $30\ \text{ €}$, $600$. The figure shows the data and the interpolating parabola.

1. a) Find the parabola $y = ax^{2} + bx + c$ that interpolates the three points (price, tickets sold).

1. b) Estimate how many tickets it would sell at $25\ \text{ €}$.

1. c) According to the model, which price maximises the number of tickets sold? Calculate the vertex of the parabola and interpret the result.

The table shows the first values of a sequence:

• $n$: $1$, $2$, $3$, $4$, $5$

• $a_n$: $4$, $9$, $18$, $31$, $48$

1. a) Calculate the first differences (each term minus the previous one) and the second differences (differences of the differences). What do you observe in the second differences?

1. b) Justify why if the second differences are constant, the sequence follows a quadratic model $a_n = an^{2} + bn + c$, and find that model.

1. c) Calculate $a_6$ using the model and check it by extending the differences.

A self-employed professional in Madrid analyzes the monthly cost of two electric carsharing companies to travel to meetings:

• The company EcoCar bills according to a linear function: for a monthly usage of $40\text{ minutes}$ it charges $18\text{ \euro}$, and for $100\text{ minutes}$ it charges $33\text{ \euro}$.

• The company UrbanMove has a rate that includes a fixed maintenance fee of $6\text{ \euro}$ per month plus a variable cost of $0,30\text{ \euro}$ per minute of driving.

1. a) Using linear interpolation with the two known records, determine the monthly cost function $C_1(x)$ of EcoCar in terms of the minutes driven $x$.
2. b) Write the analytical expression of the cost function $C_2(x)$ of UrbanMove.
3. c) Calculate the break-even point (the monthly minutes for which both bills are equal) and reasonably determine which company is more advantageous if you drive less time than that value and which one if you drive more time.
4. d) If you plan to drive $180\text{ minutes}$ in an extraordinary month, is calculating the cost with EcoCar's function an interpolation or an extrapolation? If EcoCar applies tiered discounts for usage beyond two hours, explain how this would affect the reliability of your linear estimate.

A weather station located in the Sierra de Guadarrama models the incident solar radiation $R(t)$ (in $\text{W/m}^2$) during a clear morning using an interpolation parabola $R(t) = at^2 + bt + c$, where $t$ represents the time in hours elapsed since sunrise ($t = 0$ corresponds to $6:00\text{ h}$ in the morning).

The following data are recorded: at $6:00\text{ h}$ ($t = 0$) the radiation is $0\text{ W/m}^2$, and at $8:00\text{ h}$ ($t = 2$) it is $360\text{ W/m}^2$. The record corresponding to $10:00\text{ h}$ ($t = 4$) was lost due to a computer failure, but it is known with certainty that the quadratic model predicts that maximum solar radiation is reached exactly at solar noon, at $12:00\text{ h}$ ($t = 6$).

1. a) Use the zero radiation condition at sunrise and the position of the vertex of the parabola at $t = 6$ to set up the equations that allow finding the coefficients $a$, $b$, and $c$.
2. b) Determine the quadratic function $R(t)$ and recover the lost data point: what solar radiation does the model predict for $10:00\text{ h}$ ($t = 4$)?
3. c) Calculate the maximum value of solar radiation that was reached at $12:00\text{ h}$.
4. d) If the function obtained is used to estimate the radiation at $20:00\text{ h}$ ($t = 14$), what numerical value is obtained? Explain why this result highlights the physical limits of quadratic extrapolation.

A sustainable courier company uses electric bicycles for its deliveries. It has been observed that the monthly maintenance cost for each bicycle depends on the kilometers traveled. If a bicycle travels $400\text{ km}$, the cost is $120\text{ €}$, whereas if it travels $1000\text{ km}$, the cost rises to $210\text{ €}$. The figure shows these two data points.

1. a) Using linear interpolation, determine the analytical expression of the function that relates the monthly cost $C(x)$ to the distance traveled $x$.

1. b) If the maximum maintenance budget per bicycle is $180\text{ €}$, what is the maximum number of kilometers it is estimated to be able to travel according to this model?

1. c) The company plans to expand and estimates that some bicycles will travel $1500\text{ km}$ per month. Calculate the cost using linear extrapolation and reason whether this estimate is reliable in the long term, based on the physical wear of the components.

The market value of a specific model of high-end smartphone depreciates linearly during the first few months after its launch. 2 months after its release, its price was $950\text{ €}$. At 8 months, its price dropped to $710\text{ €}$.

1. a) Calculate, using linear interpolation, the price of the device 5 months after its launch.

1. b) Determine the original launch price ($x = 0$) using extrapolation.

1. c) If a user decides to sell their smartphone when its value is exactly half of the original price, in which month should they put it up for sale according to this model?

A sociological study analyses the relationship between the number of hours per week spent on digital leisure and the score on an emotional well-being test (from 0 to 100). It is known that for 10 hours of leisure the score is 85 points, and for 35 hours the score drops to 60 points.

1. a) Obtain the linear interpolation line that models this situation.

1. b) A group of students has an average score of 72 points. Estimate how many hours of digital leisure they consume per week.

1. c) If a person consumes no digital leisure ($x = 0$), what score would be assigned to them? What if they consume 80 hours? Discuss the validity of the results at the extremes of the function.

An NGO working on the reforestation of a burnt area has recorded the growth of a type of pine tree. 3 years after transplanting, the average height was $1,8\text{ m}$. At 7 years, the height reached $3,4\text{ m}$.

1. a) Assuming linear growth, estimate the height of the pine trees at 5 and a half years.

1. b) The forestry team considers the pine trees to be "adults" when they exceed $5\text{ metres}$. How many years must pass from transplanting for them to reach that height according to linear extrapolation?

1. c) If the pine tree measured $40\text{ cm}$ at the time of transplanting ($t = 0$), check whether the linear model from the previous sections fits this initial data point. If not, explain what could be occurring biologically.

The gas bill for a building community has a variable charge that depends on consumption in $kWh$. In January, with a consumption of $2500\text{ kWh}$, the total bill (including a standing charge) was $310\text{ €}$. In February, consumption dropped to $1800\text{ kWh}$ and the bill was $233\text{ €}$.

1. a) Calculate the price per $kWh$ and the value of the standing charge on the bill using linear interpolation.

1. b) Estimate the bill for a consumption of $3000\text{ kWh}$.

1. c) Due to a new local authority tax, the standing charge will increase by $15\%$ and the price per $kWh$ by $5\%$. Write the new cost function and compare it graphically with the previous one.

Two quantities $X$ and $Y$ are linearly related. It is known that for $x_1 = 12$ we get $y_1 = 40$, and that for $x_2 = 20$ we get $y_2 = 30$. The figure shows these two data points and the region in which the relationship is valid.

1. a) Determine the value of $y$ for $x = 15$ using linear interpolation.

1. b) Find the value of $x$ for which $y$ vanishes ($y = 0$). ¿Pertenece ese valor a la zona de validez?

1. c) If this linear relationship is known to be valid only on the interval $x \in [10, 30]$, calculate the maximum and minimum values that the quantity $Y$ can take on that interval.

The supply and demand laws for a product have been modelled linearly using the following data collected from a market research study:

• At a price of $15\text{ €}$, $200$ units are supplied and $500$ are demanded.

• At a price of $25\text{ €}$, $400$ units are supplied and $300$ are demanded.

1. a) Calculate the supply function $O(p)$ and demand function $D(p)$ using linear interpolation.

1. b) Find the equilibrium price (where supply equals demand) and the equilibrium quantity.

1. c) If the government imposes a maximum price of $18\text{ €}$, will there be excess supply or a shortage of demand? Calculate the exact difference in units.

A marathon runner keeps a constant pace during a stretch of training. At kilometer 5 of the route, the stopwatch showed 22 minutes and 30 seconds. At kilometer 12, the elapsed time was 54 minutes. The figure shows these two data points.

1. a) Express the time $T$ (in minutes) as a function of the distance $d$ (in kilometers), assuming the pace is constant.

1. b) Calculate, by interpolation, at which kilometer the runner was when the stopwatch showed exactly 45 minutes.

1. c) If the runner started the marathon at kilometer 0, how long did it take to cover the first 500 meters? Use extrapolation and express the result in minutes and seconds.

The figure shows the points $(0, 2)$, $(1, 5)$ and $(2, 10)$, the line interpolating the first two, and the parabola interpolating all three.

1. a) Find the linear interpolation line passing through $(0, 2)$ and $(1, 5)$.

1. b) Find the parabola $y = ax^{2} + bx + c$ passing through all three points.

1. c) Calculate the prediction of each model for $x = 3$ and their difference. If the data were the amount of an electricity bill that grows faster and faster across consumption brackets, which model seems more reliable to you? Reason your answer.

Prove that through three points with distinct x-coordinates $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$ passes a unique polynomial function of degree less than or equal to $2$.

1. a) Set up the system of three equations with unknowns $a$, $b$ and $c$ resulting from requiring $y = ax^{2} + bx + c$ to pass through the three points.

1. b) Justify that the system has a unique solution by calculating its determinant (or discussing the system) and using the fact that the x-coordinates are distinct.

1. c) Explain why "degree less than or equal to $2$" includes the possibility of the solution being a straight line: what condition would the three points then satisfy?

On a summer day, temperatures in a city were measured: at $8$ h, $12$ °C; at $14$ h, $24$ °C; at $20$ h, $18$ °C.

1. a) Taking $t = 0$ at $8$ h, find the parabola $T(t) = at^{2} + bt + c$ interpolating the three data points.

1. b) According to the model, at what time was the maximum temperature reached, and what was its value?

1. c) What temperature does the model predict for $2$ h the following morning ($t = 18$)? Is it reasonable? Comment on the limitations of interpolation when used to extrapolate outside the data range.

The sustainable mobility yearbook records the cumulative registrations of $100\text{ \%}$ electric passenger cars (in thousands of vehicles) in an autonomous community over three periods: in 2018 ($t = 0$) there were $12$ thousand vehicles; in 2020 ($t = 2$), $20$ thousand vehicles; and in 2022 ($t = 4$), $36$ thousand vehicles. The goal is to study the evolution of the vehicle fleet to estimate the figures for 2021 ($t = 3$) and 2024 ($t = 6$).

1. a) Piecewise linear method: calculate the interpolation line connecting only the two closest available points ($t = 2$ and $t = 4$) and estimate the number of registrations in 2021 ($t = 3$).
2. b) Global quadratic method: set up the system of equations to find the interpolation parabola $P(t) = at^2 + bt + c$ that passes through the three known points ($t = 0$, $t = 2$, and $t = 4$). Find the estimate for 2021 ($t = 3$) with this model.
3. c) Compare both results for 2021 ($t = 3$). Why is the linear estimate higher than the quadratic one at that point? Taking into account that the transition to electric vehicles exhibits progressive acceleration in its early phases, justify which of the two methods better describes the trend between 2020 and 2022.
4. d) Use both models to extrapolate the number of registrations in 2024 ($t = 6$). Discuss which of the two methods seems more credible for the short term ($2024$) and argue what problems both models will present if you intend to project demand over the very long term (for example, looking $15$ years ahead).

A Spanish textile company is studying two electricity supply contracts to buffer the impact of the cost of the megawatt-hour ($\text{MWh}$) on the wholesale market (pool price, $p$, expressed in $\text{\euro/MWh}$):

• Indexed Contract: bills monthly according to a linear law $C_I(p) = mp + n$. From previous bills, the company knows that in a month with an average pool price of $50\text{ \euro/MWh}$ the bill was $2400\text{ \euro}$, while in a month with $90\text{ \euro/MWh}$ it rose to $3200\text{ \euro}$.

• Hedged Contract: offers a guaranteed flat rate of $3200\text{ \euro}$ per month as long as the pool price does not exceed $80\text{ \euro/MWh}$. If the pool exceeds that threshold ($p > 80$), the supplier applies a linear surcharge; it is known that if the pool reaches $120\text{ \euro/MWh}$, the resulting bill is $3600\text{ \euro}$.

1. a) Obtain through linear interpolation the expression of the function $C_I(p)$ for the Indexed Contract.
2. b) Determine the cost function $C_C(p)$ of the Hedged Contract for prices $p > 80\text{ \euro/MWh}$, interpolating the line that connects the hedging threshold ($p = 80$, $C = 3200$) with the recorded data point at $p = 120\text{ \euro/MWh}$.
3. c) Calculate the break-even pool price ($p > 80$) at which both contracts have the identical cost. Interpret in detail for what range of wholesale prices it is more advantageous for the textile company to sign the hedged modality.
4. d) During a month of exceptional energy crisis, the average pool price escalates to $180\text{ \euro/MWh}$. Calculate the bill predicted by both models. Justify whether this calculation represents an interpolation or an extrapolation, and critically analyze what risks company management takes if it budgets its costs by blindly relying on these linear models in extreme scenarios.