Inverse proportionality function

A group of friends wants to book a padel court that costs $20$ pesos in total. The price each person must pay depends on the number of people playing. Look at the following table, which relates the number of friends ($x$) to the amount each person pays ($y$):

Calculate the product of the two quantities ($x \times y$) for each column and verify that it is always the same. What is this constant value called in inverse proportion functions?

State which of the following algebraic expressions corresponds to an inverse proportion function and explain why:

1. a) $y = 4x$
2. b) $y = \frac{4}{x}$
3. c) $y = x + 4$
4. d) $y = \frac{x}{4}$

A group of friends wants to book a padel court for the weekend. The total price of the booking is fixed, so the cost per person depends on how many people end up playing. Look at the following table:

1. a) Check, giving reasons, whether this is an inverse proportion function.
2. b) Find the inverse proportionality constant $k$ and explain what it represents in this context.
3. c) Write the equation of the function relating both quantities.

Complete the following table of values, knowing that the inverse proportion function has a constant of $k = 12$. The formula of the function is $y = \frac{12}{x}$.

Check whether the point $A(2, 9)$ belongs to the inverse proportion function $y = \frac{18}{x}$. Explain what operation you performed to find out.

Find the inverse proportionality constant $k$ of a function, knowing that its graph passes through the point $P(3, 10)$. Once you have $k$, write the equation of the function.

(Hint: remember that in this type of function $k = x \times y$).

Look at the following graph. Without carrying out any calculations, identify whether it represents a linear function (direct proportion) or an inverse proportion function (hyperbola). Justify your answer based on the shape of the line.

A courier company has to deliver $60$ parcels. If it hires more delivery drivers, each one will have to carry fewer parcels. The relationship is given by the function $y = \frac{60}{x}$, where $x$ is the number of drivers and $y$ is the number of parcels per driver.

If the company decides to hire $6$ delivery drivers, how many parcels will each one carry?

We want to design a rectangular vegetable garden in the school playground that always has a fixed area of $24$ $m^2$.

1. a) Write the function expressing the height ($y$) in terms of the base length ($x$).
2. b) State whether the function is increasing or decreasing and justify your answer based on the context of the problem.
3. c) Plot the function graphically for base values between $1$ and $12$ meters.

Given the following graph, which represents an inverse proportionality function $y = \dfrac{k}{x}$:

1. a) Using the highlighted point $P(2, 12)$, calculate the constant $k$ of the function.

1. b) Write the algebraic expression of the function.

1. c) What would the value of $y$ be when $x = 6$? And the value of $x$ when $y = 3$?

Given the inverse proportion function $y = \frac{18}{x}$:

1. a) State its domain (bearing in mind that it is a pure mathematical function, not applied to a real-world context).
2. b) Calculate the outputs for $x = -3$, $x = 1$, $x = 2$ and $x = 9$.
3. c) Determine the points on the graph where the y-coordinate is equal to the x-coordinate ($y = x$).

Look at the following graph, which represents an inverse proportion function of the form $y = \frac{k}{x}$. Using the visual information:

1. a) Find the value of the constant $k$ knowing that the curve passes exactly through the point highlighted in orange.
2. b) Calculate the value of the y-coordinate $y$ when the x-coordinate is $x = -2.5$.
3. c) Does the point $P(12, -0.5)$ belong to the function? Justify your answer.

We want to build rectangles that all have the same area: $24$ $cm^2$. The relationship between the base ($x$) and the height ($y$) of these rectangles follows an inverse proportionality function. In the figure you can see the graph of this relationship and, as an example, a rectangle with base $6$ $cm$ and height $4$ $cm$.

1. a) Write the equation that relates the height as a function of the base.
2. b) If the base measures $8$ $cm$, what will its height be?

An electric scooter has to cover a fixed distance of $12$ km to travel from the middle school to the municipal sports center. The time taken ($y$, in hours) depends on the constant speed at which it travels ($x$, in km/h).

1. a) Complete the table of values relating speed and time:

1. b) Write the equation of the function.
2. c) If the municipality limits the speed of electric scooters to a maximum of $25$ km/h, what is the minimum time (in minutes) it will take to complete the trip?

State which of the following situations can be modelled by an inverse proportion function and write its equation in that case:

1. The number of tickets for a concert that we can buy with $2\,000$ $\$$ depending on the price of each ticket.

2. The distance covered by a car travelling at a constant speed of $90$ km/h depending on the elapsed time.

3. The number of workers needed to build a wall depending on the days it takes to finish it (working at the same rate).

4. The price of a bag of oranges costing $30$ $\$$ per kilo.

A tanker truck with a capacity of $15,000$ liters is emptied using several identical extraction pumps. If $2$ pumps are used, the tanker is emptied in $6$ hours.

1. a) How long would it take if $4$ pumps were used?
2. b) And if we wanted to empty it in only $2$ hours, how many pumps would we need?
3. c) Identify the independent and dependent variables and write the formula of the function relating them.

A group of 2nd-year ESO students has won a shared prize of $600$ $\$$ in a robotics competition. The prize is split equally among all the members of the team.

1. a) Build a table that relates the number of students on the team ($x$) to the amount of money each one receives ($y$).

1. b) Graph the function. Is it a continuous or a discontinuous function? Justify your answer.

1. c) If each student wants to receive at least $75$ $\$$, what is the maximum number of students that can be on the team?

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

In an animal sanctuary, a batch of feed has been bought that lasts exactly $24$ days to feed the $80$ wolves in the enclosure. Due to a rescue, the number of wolves is going to change.

1. a) Determine the inverse proportionality constant $k$ and explain what it represents in this context.
2. b) Express the function relating the number of days $d$ the feed will last to the number of wolves $n$.
3. c) If, after the rescue, the feed only lasts $15$ days, how many new wolves have arrived at the sanctuary?

A group of friends wants to rent a vacation home for the December long weekend, which costs $14\,400$ pesos in total. The price per person depends on the number of friends who end up going on the trip.

1. a) Write the equation of the function.
2. b) If there were initially $6$ friends, but $3$ more people join, how much money does each of the initial $6$ friends save?
3. c) If each friend's maximum budget is $1\,000$ pesos, what is the minimum number of people needed to rent the house?

Consider a rectangle whose area is constant and equal to $48$ cm². The base of the rectangle is $x$ and the height is $y$.

1. a) Express the height as a function of the base. What type of function is it?

1. b) If we increase the base by $4$ cm, the new height is $4$ cm. What was the length of the original base?

1. c) Graph the situation for base values between $2$ and $16$ cm.

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

Two quantities $A$ and $B$ are inversely proportional. It is known that when $A$ increases by $5$ units, the value of $B$ is halved from its original value (which was $B = 20$).

1. a) Calculate the initial value of quantity $A$.
2. b) Determine the inverse proportionality constant $k$.
3. c) Write the algebraic expression of the function relating both quantities.

A gear system is made up of two cogwheels, $R_1$ and $R_2$. The number of teeth and the rotation speed (in rpm) are inversely proportional quantities. Wheel $R_1$ has $40$ teeth and rotates at $120$ rpm.

1. a) If wheel $R_2$ rotates at $300$ rpm, how many teeth does it have?
2. b) If we want the second wheel to rotate as slowly as possible, but it must have at most $80$ teeth, what will its minimum rotation speed be?
3. c) Find the function that gives the rotation speed $y$ in terms of the number of teeth $x$.

Given the following table of values of an inverse proportion function, find the hidden values marked with letters and answer the questions:

1. a) Calculate the values of $a$, $b$, $c$ and $d$.
2. b) State whether the function is increasing or decreasing in the first quadrant.
3. c) For what value of $x$ is an image of $y = 0.01$ obtained?