Inverse proportionality function
If you make a 240 km journey at 60 km/h it takes 4 hours; at 120 km/h, only 2. The greater the speed, the less the time, and always in the same proportion: speed $\cdot$ time = 240. You already know this relationship from inverse proportion. Now you are going to see its graph.
The function $y = \frac{k}{x}$
A server shares $k$ GB of download among several devices at the same time, each at its own speed. Change $k$:
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The greater $k$ is, the "higher" the curve sits: with the same sharing, each one gets more.
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If you know a point $(x, y)$ of the function, you already know $k$: $k = x \cdot y$.
A group of friends wants to book a padel court that costs $20$ € in total. The price each person must pay depends on the number of people playing. Look at the following…
A group of friends wants to book a padel court that costs $20$ € 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 ($y$):
| Number of friends ($x$) | $1$ | $2$ | $4$ | $5$ |
|---|---|---|---|---|
| Price per person ($y$) | $20$ | $10$ | $5$ | $4$ |
Calculate the product of the two quantities ($x \cdot y$) for each column and verify that it is always the same. What is this constant value called in inverse proportion functions?
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…
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:
| Number of friends ($x$) | 2 | 3 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| Cost per person ($y$) in $€$ | $12$ | $8$ | $6$ | $4$ | $3$ |
- a) Check, giving reasons, whether this is an inverse proportion function.
- b) Find the inverse proportionality constant $k$ and explain what it represents in this context.
- c) Write the equation of the function relating both quantities.
The graph: the hyperbola
The graph of $y = \frac{k}{x}$ (with $k > 0$ and $x > 0$, the most common case in problems) has this shape, getting closer and closer to the axes without ever touching them:
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The function is decreasing: the greater $x$ is, the smaller $y$ is.
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If in your problem $x$ can only be a natural number (people, wolves, tickets), the real graph is separate points on the curve, not a continuous 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…
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 yard that always has a fixed area of $24\ \text{m}^2$. a) Write the function that expresses the length of…
We want to design a rectangular vegetable garden in the school yard that always has a fixed area of $24\ \text{m}^2$.
- a) Write the function that expresses the length of the height ($y$) in terms of the length of the base ($x$).
- b) State whether the function is increasing or decreasing and justify your answer based on the context of the problem.
- c) Graph the function for values of the base between $1$ and $12$ meters. You can use a table of values to help you.
You can do it on paper and send a photo with your answer.
Asymptotes and the two branches
If we also let $x$ take negative values, $y = \frac{k}{x}$ has two branches, and the curve gets closer to both axes without ever touching them: these are the asymptotes.
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$k > 0$: the branches lie in quadrants I and III.
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$k < 0$: the branches lie in quadrants II and IV, and the function is increasing on each branch.
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It never crosses either axis: $y = \frac{8}{x}$ is never 0, and it does not exist at $x = 0$.
Compare the two signs, leaving aside the case $k = 0$, which is not a hyperbola:
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. a) Build a…
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.
- a) Build a table that relates the number of students on the team ($x$) to the amount of money each one receives ($y$).
- b) Graph the function. Is it a continuous or a discontinuous function? Justify your answer.
- 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.
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…
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 $B = 20$).
- a) Calculate the initial value of quantity $A$.
- b) Determine the inverse proportionality constant $k$.
- c) Write the algebraic expression of the function relating both quantities.
Finding $k$ and the complete formula
With just one data point (a pair $x, y$) you can already write the whole function.
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…
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 $4$ $cm$.
- a) Write the equation that relates the height as a function of the base.
- b) If the base measures $8$ $cm$, what will its height be?
A tanker lorry with a capacity of $15.000$ litres is emptied using several identical extraction pumps. If $2$ pumps are used, the tanker is emptied in $6$ hours. a) How…
A tanker lorry with a capacity of $15.000$ litres is emptied using several identical extraction pumps. If $2$ pumps are used, the tanker is emptied in $6$ hours.
- a) How long would it take if $4$ pumps were used?
- b) And if we wanted to empty it in only $2$ hours, how many pumps would we need?
- c) Identify the independent and dependent variables and write the formula of the function relating them.
A video game server has a fixed data transfer capacity to download an update of $1,2$ GB ($1228,8$ MB). The download speed $v$ (in MB/s) and the required time $t$ (in…
A video game server has a fixed data transfer capacity to download an update of $1,2$ GB ($1228,8$ MB). The download speed $v$ (in MB/s) and the required time $t$ (in seconds) are inversely proportional quantities.
- a) Write the function that relates the time $t$ to the speed $v$.
- b) If a user has a connection limited to $4$ MB/s, how many minutes will it take to download the update?
- c) If we want the download to take no more than $2$ minutes, what must the minimum speed of our connection be?