Studying Lines

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 $(0, 0)$?

Renting an electric scooter costs $\$20$ as an initial unlock fee and $\$10$ for each minute of use. The function relating the cost ($y$) to the time in minutes ($x$) is $y = 10x + 20$.

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

A new music streaming service offers a subscription costing $\$60$ for each month purchased.

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

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

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 bicycle parking facility in the city center charges a flat rate of $\$40$ 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?

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.

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

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).

Given the following graph representing an affine function, complete the following steps:

1. a) Visually identify the y-intercept ($n$).
2. b) Calculate the slope ($m$) using two points on the line that it passes through exactly (for example, the intercepts with the axes).
3. 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.

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.

A drone is initially at a height of $40$ meters. Upon pressing the "return to base" button, the drone begins to descend at a constant speed of $4$ meters per second.

1. a) Define the independent and dependent variables in this problem.
2. b) Write down the equation of the function that relates the height of the drone ($y$) to the time elapsed ($x$) since the button was pressed.
3. c) What is the height of the drone after $6$ seconds? How long does it take to land completely (height $0$)?

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?

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

a) Find the equation of the linear function relating the elapsed time ($x$) to the volume of water in the tank ($y$).

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

c) Plot the function and state the domain and range of the function, taking into account the physical context of the problem (time cannot be negative and volume cannot be less than zero).

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.

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.

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.

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 electric scooter rental companies have different rates. EcoRolling charges $\$4$ per minute of use with no fixed fee. FastGlide charges a fixed unlocking fee of $\$40$ and then $\$2$ per minute.

1. a) Write the function relating total cost ($y$) to usage time in minutes ($x$) for each company.
2. b) State which of the two is a direct proportion (linear) function and which is an affine function.
3. c) Above how many minutes of travel does FastGlide start to be the cheaper option?

Two fiber optic companies offer the following monthly tariffs:

• NetRápido: a fixed $400$ pesos per month plus $10$ pesos for each extra GB downloaded.

• FibraTotal: no fixed fee, but charges $26$ pesos 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?

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?

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 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.