Exercises: Linear functions. Gradient

Find the gradient of the line passing through the points $A(1, 4)$ and $B(3, 10)$.

Given the linear function $y = -3x + 5$, state its gradient and y-intercept. Is it an increasing or a decreasing function?

Look at the following graph and determine the gradient of the line shown.

Write the explicit equation of the line that has a gradient of $m = \frac{1}{2}$ and intersects the y-axis at the point $(0, -4)$.

A linear relationship between two variables is given by the following table of values:

Calculate the gradient of this function.

A tap fills a tank at a rate of $3$ litres per minute. If there were initially $10$ litres in the tank, what is the gradient of the function relating the volume of water to the time elapsed?

Given the following lines: $y = 2x$, $y = 0,5x$, $y = -x$, $y = -4x$. Order them from least to steepest gradient (absolute value of their gradient) and classify them into increasing and decreasing.

The line passing through the points $P(2, 5)$ and $Q(4, y)$ has a gradient of $m = 3$. Calculate the value of the unknown coordinate $y$.

Marcos has to calculate the slope of the line passing through points $A(2, 3)$ and $B(6, 11)$. On his exam, he writes the following procedure:

Therefore, he concludes that for every unit the line rises vertically, it moves $0,5$ units horizontally.

1. a) Point out precisely what conceptual error Marcos made when applying the slope formula.
2. b) Explain why his subsequent verbal interpretation is also incorrect according to the concept of slope as a rate of change.
3. c) Correct the calculation to find the right value of $m$ and reason whether the function is increasing or decreasing based on its sign.

In the taxi service of a Spanish provincial capital regulated by official urban fares, the base fare costs €$2,40$ (a mandatory fixed amount for starting the trip) and the price for each kilometre travelled is €$1,25$.

1. a) Write the equation of the linear function $y = mx + n$ that relates the total cost of the trip ($y$, in euros) to the kilometres travelled ($x$).
2. b) Identify the value of the slope $m$ and write a sentence explaining the physical and economic meaning of this number in the context of the trip.
3. c) A passenger claims: "If for a trip of $4$ km I paid €$7,40$, for travelling twice as far ($8$ km) I will pay exactly twice as much, €$14,80$". Use the slope and the $y$-intercept to mathematically justify why this statement is false.

Calculate the gradient of the line passing through the points $A(-2,5; 3,2)$ and $B(1,5; -4,8)$.

An aircraft descends at a constant rate to land. $10$ minutes after starting the manoeuvre, it is at an altitude of $8000$ metres, and at $25$ minutes, it is at $3500$ metres. Calculate the gradient of the function relating altitude to time and interpret its meaning in this context.

Determine the gradient of the line whose general equation is $4x - 5y + 20 = 0$. Next, write its equation in explicit form.

Observe the two lines shown in the graph. Calculate the gradient of each line and state which one represents a higher rate of growth.

A bicycle hire company charges a fixed initial fee plus a rate per hour of use. If hiring a bicycle for $2$ hours costs $14$ € and hiring it for $5$ hours costs $26$ €, find the gradient of the cost function and explain what this value represents in the context of the problem.

Find the value of the parameter $k$ so that the line passing through the points $A(k, 3)$ and $B(2k - 1, 7)$ has a gradient of $m = -2$.

Two quantities $x$ and $y$ have a linear relationship. If we know that when $x$ increases by $5$ units, $y$ decreases by $3$ units, and that the line passes through the point $(10, -2)$, write the explicit equation of the linear function relating them.

Write the equation of a line parallel to $y = -\frac{3}{4}x + 2$ passing through the point $(-8, 5)$.

According to meteorological records from the State Meteorological Agency (AEMET) in a mountainous area of the peninsula, at an altitude of $500$ m above sea level a temperature of $19$ ºC is recorded, while at a station located at $2000$ m the temperature drops to $10$ ºC.

1. a) Before performing formal calculations, reasonably estimate the sign of the slope and predict whether the temperature will drop more or less than $1$ ºC for every $100$ m of ascent.
2. b) Assuming a linear variation, calculate the slope $m$ of the function expressing temperature ($y$, in ºC) in terms of altitude ($x$, in metres).
3. c) Check whether your initial estimate was correct, express the slope as a change per $100$ m, and interpret what it would physically mean in the atmosphere if the slope were $m = 0$.

It is desired to design a line $r$ in the Cartesian plane with a fixed slope $m = -1,5$, but several of its initial elements are unknown.

1. a) Knowing that line $r$ must pass through point $A(2, 4)$, carry out the inverse procedure to determine its $y$-intercept $n$ and write its explicit equation $y = mx + n$.
2. b) Set up and solve an equation based on the definition of slope to find the unknown horizontal coordinate of a point $B(x_B, -2)$ belonging to this line.
3. c) Write the general equation of another line $s$ that is parallel to $r$ and whose $y$-intercept is three times that of $r$, justifying how you use the parallelism condition.

A spring stretches proportionally to the applied force (Hooke's Law). Hanging a mass of $2$ kg gives a total spring length of $15$ cm. Hanging a mass of $5$ kg gives a total length of $21$ cm. Calculate the gradient of the function relating length to weight, write its general equation, and state the length of the spring at rest (with no weight).

Check algebraically whether the points $A(-2, 5)$, $B(1, -1)$ and $C(5, -9)$ are collinear by calculating and comparing the gradients of the line segments joining them.

The following graph shows the journey of a freight train making an outward trip, making a stop, and returning to its point of origin. Calculate the gradient of each of the three sections and interpret the result as speed (in km/h).

The value of an industrial machine depreciates linearly over time. It was bought new for $45000$ € and, after $6$ years, it has been valued at $18000$ €. Calculate the gradient of the value-time function, determine in which year the machine will theoretically lose all its value, and write the algebraic expression for the function.

Given the lines $r: ax + 3y - 6 = 0$ and $s: 2x - by + 4 = 0$, determine the values of the parameters $a$ and $b$, knowing that line $r$ has a gradient of $m = 2$ and that both lines are strictly parallel.

Prove algebraically that the Average Rate of Change (A.R.O.C.) of the linear function $f(x) = mx + n$ in any generic interval $[a, b]$ is always constant and equal to its gradient $m$.

A line with gradient $m = -\frac{2}{3}$ forms a right-angled triangle with the coordinate axes in the first quadrant. If the area of this triangle is $27$ square units, find the explicit equation of the line.

An industrial tank contains $1200$ litres of water and begins to drain at a constant rate. If the gradient of the function relating the volume of remaining water (in litres) to the elapsed time (in hours) is $m = -40$, find how long it will take for the tank to reach exactly one quarter of its initial maximum capacity.

To check whether three agricultural plots whose access points are located on a cadastral map at points $A(-2, -5)$, $B(2, 1)$, and $C(6, 7)$ are perfectly aligned along a straight irrigation canal, two technicians propose different methods:

• Laura's method: Calculate the slope of the segment between $A$ and $B$, then the slope of the segment between $B$ and $C$, and check if they match.

• Carlos's method: Find the explicit equation of the line connecting $A$ and $B$ ($y = mx + n$) and then check whether the coordinates of $C$ satisfy this equation.

1. a) Solve the problem step by step using Laura's method and determine whether the points are aligned.
2. b) Solve the same problem using Carlos's method to confirm the result.
3. c) Analyse both procedures: which of the two methods is more efficient and convenient if a surveyor needed to check the alignment of a series of ten consecutive plots? Justify your choice by assessing the computational workload of each option.

According to reports from the Directorate-General for Water of the Ministry for the Ecological Transition and the Demographic Challenge (MITECO), at the beginning of the summer (week $0$) a reservoir in a southern peninsular river basin held $360$ hm³ of water. After $15$ continuous weeks of heat without rain, the volume decreased steadily to $210$ hm³ due to consumption and evaporation.

1. a) Model the evolution of the water reserve ($y$, in hm³) as a function of the weeks elapsed ($x$) using a linear function. Calculate the slope $m$ and interpret its sign and units in terms of the weekly rate of change.
2. b) The drought protocol declares the emergency phase when the reservoir drops below $100$ hm³. Determine with justification in which week this phase would be reached if conditions remain unchanged.
3. c) In the same river basin, a second neighbouring reservoir had a volume of $270$ hm³ in week $0$. Knowing that the line describing its emptying is parallel to that of the first reservoir, explain what practical meaning this parallelism has regarding its rate of water loss and reason, without needing to calculate the complete equation of the second reservoir, which of the two will enter the emergency phase first.