Exercises: Study of straight lines

A group of friends is designing a new logo for their gaming club on grid paper. Write down the coordinates of the key points shown on the graph that form the vertices of the design.

For a graph to represent a function, each value of the independent variable $x$ must correspond to a single value of the dependent variable $y$. Look at the following graphs and state, giving reasons, which of them is not a function. To justify your answer, imagine vertical lines sweeping across each graph and notice how many points they cross it at.

A new music streaming service offers a subscription costing $3 \text{ €}$ for each month purchased.

• a) Complete the following table of values relating the time in months ($x$) to the total cost ($y$):

Months ($x$)	1	2	3	4	5
Cost (€) ($y$)	$3$

• b) Write down the equation of the direct proportion function that represents this situation.

Given the linear function $y = -2x$, state with reasons:

• a) What is the value of its gradient ($m$)?

• b) Is the function increasing or decreasing?

• c) Does the graph pass through the origin $(0, 0)$?

Find the gradient ($m$) and the y-intercept ($n$) of the line shown in the following graph. Then, write its equation in the form $y = mx + n$.

A bicycle parking facility in the city centre charges a flat rate of $2 \text{ €}$ per day, regardless of how many hours the bicycle is parked.

• a) Represent this situation using a table of values (hours vs cost).

• b) What type of function is it: linear, affine, or constant?

• c) What would its equation be?

Renting an electric scooter costs $1 \text{ €}$ as an initial unlock fee and $0,5 \text{ €}$ for each minute of use. The function relating the cost ($y$) to the time in minutes ($x$) is $y = 0,5x + 1$.

Calculate how much a trip lasting $10$ minutes will cost.

Look at the following graph showing the journey of a person walking at a constant speed. State whether the relationship between the distance covered and the time is directly proportional, and calculate its constant of proportionality (the gradient).

Two electric scooter rental companies have different rates. The company EcoRolling charges $0,20\ €$ for each minute of use with no fixed fee. The company FastGlide charges a fixed fee of $2\ €$ for unlocking the scooter and then $0,10\ €$ for each minute. The graph shows both rates.

1. a) Write the function that relates the total cost ($y$) to the usage time in minutes ($x$) for each of the companies.

1. b) State which of the two is a direct proportionality (linear) function and which is an affine function.

1. c) After how many minutes of travel does the FastGlide option start to be cheaper?

Look at the graph of an affine function. Each square of the grid measures 1 unit on each axis.

Carry out the following steps:

1. a) Visually identify the y-intercept ($n$), that is, the value of $y$ at the point where the line crosses the vertical axis.

1. b) Calculate the slope ($m$) using two points on the line that lie exactly on grid crossings (the points marked in orange can help you).

1. c) Write the explicit equation of the line in the form $y = mx + n$.

Find the equation of the line passing through points $A(-2, 5)$ and $B(4, -7)$. To do this:

1. First, calculate the gradient $m$ using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.

2. Substitute the gradient and one of the points into the equation $y = mx + n$ to find the value of $n$.

3. Write down the final equation and state whether the function is increasing or decreasing, justifying your answer based on the value of the gradient.

A drone is initially at a height of $40$ meters. When the "return to base" button is pressed, the drone starts to descend at a constant speed of $4$ meters per second. The figure shows only the beginning of the descent.

1. a) Define the independent and dependent variables of this problem.

1. b) Write the equation of the function that relates the drone's height ($y$) to the time elapsed ($x$) since the button was pressed.

1. c) What is the drone's height after $6$ seconds? How long does it take to land completely (height $0$)?

Determine the equation of the line that satisfies the following two conditions:

• It is parallel to the line $y = -3x + 5$.

• It passes through the point $P(0, 4)$.

Explain what property two lines must share to be parallel and how you have used the point $P$ to find the y-intercept.

Match each of the following equations of lines with its corresponding description:

1. $y = 3$

2. $x = -2$

3. $y = 3x$

4. $y = -x + 3$

A) Line passing through the origin with a positive gradient.

B) Constant function representing a horizontal line.

C) Line with a negative gradient and a y-intercept equal to $3$.

D) Vertical line that does not represent a function.

Kenia is saving money in a piggy bank. Initially, she has $15\ €$, and each week she decides to put in $2,50\ €$ from her pocket money.

1. a) Complete the following table of values:

Weeks ($x$)	$0$	$1$	$2$	$4$	$10$
Money saved ($y$)	$15$

1. b) Write down the equation of the function.
2. c) If Kenia wants to buy a video game costing $55\ €$, how many weeks will she have to wait according to this function?

Draw on the same coordinate axes the graphs of the functions $y = 2x - 1$ and $y = 2x + 3$.

1. a) What geometric relationship do you observe between the two lines?

1. b) Justify your previous answer by comparing their equations.

1. c) Will these two lines intersect at any point? Explain your reasoning.

You can do it on paper and send a photo with your answer.

A digital content creator analysed the growth of their followers. During the first month, their growth followed a linear function passing through points $A(2, 500)$ and $B(5, 1250)$, where $x$ is the day of the month and $y$ is the number of followers. However, from day 15 onwards, due to a viral video, their growth became another linear function parallel to the previous one, but which on day $x = 16$ had already reached $5000$ followers.

a) Find the equation of the linear function for the first section (days 1 to 15).

b) Find the equation of the new function from day 15 onwards.

c) How many followers did they have on day 20 according to this second function?

In a drone competition, "Drone A" follows a straight-line trajectory represented by the function $y = \frac{3}{2}x + 2$. A "Drone B" is to be launched whose trajectory is a straight line parallel to that of "Drone A", but must pass through the point of a beacon located at the coordinates $P(-4, -1)$.

a) Determine the equation of the linear function describing the trajectory of "Drone B".

b) At what point will the trajectory of "Drone B" cross the y-axis (axis $Y$)?

c) Plot both trajectories on a graph and state whether either of them is a direct proportion function.

You can do it on paper and send a photo with your answer.

Two fibre optic companies offer the following monthly tariffs:

• NetRápido: a fixed $20$ € per month plus $0,50$ € for each extra GB downloaded.

• FibraTotal: no fixed fee, but charges $1,30$ € for each GB downloaded.

a) Write the algebraic expression of the function relating monthly cost ($y$) to extra GB ($x$) for each company.

b) Above how many extra downloaded GB does NetRápido start to be better value? (Find the point of intersection of both lines).

c) If in one month you use $30$ extra GB, what is the price difference between the two?

Find the equation of the line $r$ that satisfies the following conditions simultaneously:

1. It is decreasing.

2. Its slope is twice (in absolute value) the slope of the line that passes through the points $C(0, 1)$ and $D(2, 2)$.

3. It passes through the intersection point of the lines $y = 2x - 5$ and $y = 3$.

a) First calculate the slope of the line that passes through $C$ and $D$.

b) Determine the intersection point mentioned in condition 3.

c) Write the final equation of the line $r$.

A laptop battery discharges linearly. At 10:00 in the morning, the battery was at $90\%$. At 12:30 in the morning, the level had dropped to $45\%$.

a) If we let $x$ be the time elapsed in hours since 10:00, find the equation of the function giving the percentage of remaining battery.

b) What is the gradient of the function? Explain its meaning in this real-life context.

c) At what exact time will the battery run out completely if consumption remains the same?

It is known that a linear function is represented graphically by a line passing through the point $A(3, -2)$. Furthermore, we know that this line forms a triangle with the coordinate axes whose area is $4$ square units, and its y-intercept is a positive integer.

a) Determine the value of the y-intercept ($n$).

b) Calculate the gradient ($m$) of the line.

c) Write the equation of the line and check that the point $A$ belongs to it.

Given the functions $f(x) = ax + 3$ and $g(x) = -2x + b$:

a) If both lines are parallel and line $g(x)$ passes through the point $P(1, 4)$, calculate the values of $a$ and $b$.

b) Once $a$ and $b$ have been found, calculate the point where line $f(x)$ crosses the x-axis (axis $X$).

c) Is either of these functions a constant function? Justify your answer.

A water tank of $1000$ liters has a leak. Every hour that passes, the volume of water decreases by a constant amount. $3$ hours after the leak is detected, $850$ liters remain.

a) Find the equation of the affine function that relates the elapsed time ($x$) to the volume of water in the tank ($y$).

b) If the leak is not repaired, how much water will be left after $10$ hours?

c) Graph the function on axes like those in the figure and state the domain and range of the function taking into account the physical context of the problem (time cannot be negative and the volume cannot be less than zero).

You can do it on paper and send a photo with your answer.