Exercises: Concept of a linear function. Types

Identify the gradient ($m$) and y-intercept ($n$) of the following linear functions:

• a) $y = 5x - 2$

• b) $y = -\frac{3}{4}x + 1$

• c) $y = 8$

• d) $y = \frac{x}{2}$

Classify the following functions into direct proportion functions (linear functions), linear functions (affine functions), or constant functions:

• a) $y = -3x + 5$

• b) $y = \frac{4}{5}x$

• c) $y = -7$

• d) $y = x$

Write the equation of the direct proportion function whose graph passes through the point $P(4, -12)$.

Given the linear function $f(x) = -2x + 9$, calculate the images of $x = 3$, $x = -1$ and $x = \frac{1}{2}$.

Write the equation of the constant function passing through the point $A(-3, 6)$. What is the exact value of its gradient?

Determine mathematically whether the points $A(2, 5)$ and $B(-1, -3)$ lie on the graph of the function $y = 3x - 1$.

Calculate the gradient of the line passing through the points $A(0, 4)$ and $B(2, 10)$.

A direct proportion function is given by the following table of values. Find its algebraic expression $y = f(x)$.

• $x$: $2, 4, 6$

• $y$: $7, 14, 21$

In the urban mobility service of a Spanish city, the electric scooter company charges a fixed $1$ € unlocking fee plus $0,20$ € for each minute of the journey. In contrast, the municipal shared bicycle service charges only $0,15$ € per minute, with no unlocking fee.

1. a) Write the function that expresses the cost $y$ (in euros) as a function of the usage time $x$ (in minutes) for each of the two means of transport.
2. b) Classify each function as linear or affine, identifying in each one the slope ($m$) and the $y$-intercept ($n$), and explain what these two values mean on the bill.
3. c) If you have a maximum budget of $3$ €, what is the maximum number of full minutes you can ride the electric scooter?

To classify the function $y = 8 - 2x$ and find its initial behavior, Carlos writes on his exam:

"It is a linear function because its graph is a straight line. Its slope is $m = 8$ because it is the first value that appears, and its $y$-intercept is $n = -2$. Furthermore, since it is linear, the graph necessarily passes through the point $(0, 0)$".

1. a) Point out the three conceptual mistakes Carlos made in his explanation.
2. b) Correct Carlos's statements by indicating the actual slope and $y$-intercept of the function, and classify it correctly, justifying your choice.
3. c) Prove numerically that the graph does not pass through the origin $(0,0)$ by calculating the image of $x = 0$.

A plumber charges a fixed call-out fee of $40$ €, plus $25$ € per hour of work. Write the function relating the total cost of the service ($y$) to the hours worked ($x$). How much will a repair that took $3,5$ hours cost?

Observe the following graph showing two lines, $A$ and $B$. Determine the algebraic expression for each of them by analysing the points they pass through.

A spring measures $15$ cm at rest. When weights are attached to it, it is found that for each additional kilogram the spring stretches by $3$ cm (according to Hooke's law). Write the linear function relating the total length of the spring ($L$) to the attached mass in kilograms ($m$). What length will it have if a mass of $4,5$ kg is attached?

Find the equation of a line parallel to the function $y = -2x + 5$ that passes through the origin.

The relationship between the temperature in degrees Celsius ($C$) and degrees Fahrenheit ($F$) is linear. We know that water freezes at $0$ ºC, which is equivalent to $32$ ºF, and boils at $100$ ºC, which is equivalent to $212$ ºF. Find the function $F(C)$ that converts degrees Celsius into degrees Fahrenheit.

Calculate the exact points of intersection with the coordinate axes ($X$-axis and $Y$-axis) of the linear function $y = \frac{2}{5}x - 4$.

A tank contains $800$ litres of water and is being drained by a pump that extracts $40$ litres per minute. Write the algebraic expression for the volume of water in the tank as a function of time. How long will it take to empty completely?

Find the equation of the linear function passing through the points $P(-2, 5)$ and $Q(1, -4)$.

According to meteorological data from AEMET, the average vertical thermal gradient in the troposphere causes the temperature to drop by approximately $0,65$ °C for every $100$ meters of elevation gain ($0,0065$ °C per meter). On a spring day at 12:00, the weather station in Granada (at $700$ m above sea level) records a temperature of $20$ °C. The goal is to predict the temperature at the Pradollano ski resort (at an altitude of $2100$ m).

1. a) Before doing the calculations, estimate with reasoning whether the temperature in Pradollano will be above or below $10$ °C by calculating mentally approximately how many degrees will be lost over the $1400$ m ascent.
2. b) Model the situation using an affine function $T(x) = mx + n$ that expresses the temperature $T$ (in °C) as a function of the altitude climbed $x$ (in meters, taking $x = 0$ in the city of Granada). State the value and meaning of the sign of the slope $m$.
3. c) Use the function to calculate the exact expected temperature in Pradollano and check whether your initial estimate was accurate and consistent with the result.

A data analyst needs to model three usage rates that intersect exactly at the same control point: the point $P(5, 20)$, where $x$ represents data consumption in gigabytes and $y$ the cost in euros. Find the requested algebraic expressions:

1. a) Determine the formula of a linear function $f(x)$ that passes through the point $P$. Justify step by step how you calculate its slope from the coordinates of $P$.
2. b) Formulate the equation of an affine function $g(x)$ that also passes through the point $P$, but whose fixed fee ($y$-intercept) is $5$ €. Calculate the slope that this line must have.
3. c) Write the expression of a constant function $h(x)$ containing the point $P$. State the value of its slope and describe what its graph looks like on the Cartesian coordinate axes.

Two phone companies offer different plans. In the graph of the monthly cost in euros ($C$) as a function of the minutes used ($t$), the line for Plan A passes through the points $(0, 10)$ and $(200, 20)$, while that of Plan B passes through the origin $(0, 0)$ and $(200, 20)$. From the graph, determine the algebraic expression of each plan and explain, by solving the corresponding system, from how many minutes it is better to choose Plan A over Plan B.

Find the value of parameter $k$ so that the line with equation $y = (3k - 2)x + 5$ is coincident with the line passing through points $A(-1, 2)$ and $B(3, 14)$.

An industrial machine is bought for $15\,000$ €. Its value depreciates linearly over time until it reaches a value of $0$ € after $12$ years. Determine the function expressing the value of the machine as a function of the years elapsed. What will its exact residual value be when it is $7$ years old?

The following graph represents a piecewise function made up of three line segments. Determine its full algebraic expression $f(x)$, clearly stating the intervals of definition for each section.

Calculate the area of the triangle formed by the line $y = -\frac{3}{2}x + 6$ intersecting the coordinate axes $X$ and $Y$.

Two ships are in two ports separated by $300$ km. Ship $A$ sets sail towards the port of ship $B$ at a constant speed of $40$ km/h. At the same time, ship $B$ sets sail towards the port of ship $A$ at $35$ km/h. Write the position functions for both ships with respect to the home port of ship $A$ and calculate algebraically the exact time at which they will cross paths.

Find the equation of the line passing through the point $P(2, 3)$ given that its y-intercept is exactly double its x-intercept (that is, the point of intersection with the $Y$-axis is double the point of intersection with the $X$-axis).

Prove mathematically that if $f(x)$ is a direct proportion function ($f(x) = mx$), it always holds that $f(a+b) = f(a) + f(b)$ for any values $a$ and $b$. Then, check and justify why this property does not hold for a linear function $g(x) = mx + n$ (with $n \neq 0$).

In the automotive laboratory of a vocational training centre, the following data is recorded about the remaining fuel in the tank of a hybrid vehicle traveling at a constant speed along a highway:

• At $100$ km travelled, $45$ liters remain.

• At $200$ km, $40$ liters remain.

• At $300$ km, $35$ liters remain.

• At $400$ km, $30$ liters remain.

Two students discuss how to classify the function that relates the kilometers ($x$) to the remaining liters ($y$):

• Alba proposes calculating the ratio $\frac{y}{x}$ at each point to check whether it is constant.

• Bruno proposes calculating the rate of change $\frac{y_2 - y_1}{x_2 - x_1}$ between consecutive pairs of points and checking the $y$-intercept.

1. a) Apply Alba's proposal. What ratios do you get? What erroneous conclusion about the type of function could be drawn if we rely solely on this method?
2. b) Apply Bruno's proposal. What constant value do you get for the slope? Write the full expression of the function $y = mx + n$.
3. c) Compare both procedures: justify which of the two methods is the only one suitable for identifying and constructing affine functions in experimental tables and why the other method only works for linear functions.

In the Joint Provision Area of the Madrid taxi service (fare 1, weekdays from 7:00 to 21:00), the base fare is $2,50$ € and each kilometer travelled costs $1,30$ €. On the other hand, a VTC vehicle company operating in the same area charges a fixed base fee of $0,70$ € and a per-kilometer rate of $1,80$ €/km for that same time slot.

1. a) Write the function for the total cost $y$ (in euros) in terms of the kilometers $x$ for each of the two transport services. Identify the slope and the $y$-intercept of both, and classify them with reasoning according to the types of functions studied.
2. b) Interpret the practical meaning of the slopes: which service makes the trip more expensive for each additional kilometer travelled? Based on this and looking at the fixed fees, reason conceptually which of the two services will be cheaper for short trips and which for long trips.
3. c) Calculate, with justification, at what exact number of kilometers the cost is identical in both services. If you have to travel from Chamartín station to Adolfo Suárez Madrid-Barajas airport (a journey of $12$ km), which option should you choose, and how many euros will you save?