Linear functions: interpolation

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.

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).

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.

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.

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.

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.

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.

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.