Inverse proportionality function
The functions studied so far —first and second degree— have domain $\mathbb{R}$. The inverse proportionality function is the first in this block whose domain is not all of $\mathbb{R}$: a division appears, and a division imposes an algebraic restriction. That restriction is the key to its whole graph.
Algebraic definition and domain
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If $k>0$: the graph lies in the first and third quadrants.
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If $k<0$: it lies in the second and fourth quadrants.
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The greater $|k|$ is, the further the curve is from the axes.
Identify in the following table whether the quantities $x$ and $y$ are inversely proportional and, if so, find the constant of proportionality $k$ and write the function…
Identify in the following table whether the quantities $x$ and $y$ are inversely proportional and, if so, find the constant of proportionality $k$ and write the function relating them.
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$x$: 1, 2, 3, 4, 6
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$y$: 24, 12, 8, 6, 4
Given the inverse proportion function $y = \frac{18}{x}$, calculate the value of $y$ for $x = 3$, $x = -9$ and $x = 0,5$.
Given the inverse proportion function $y = \frac{18}{x}$, calculate the value of $y$ for $x = 3$, $x = -9$ and $x = 0,5$.
Asymptotes and symmetry of the hyperbola
Determine the domain, the range and the equations of the asymptotes of the function $y = \frac{-8}{x}$.
Determine the domain, the range and the equations of the asymptotes of the function $y = \frac{-8}{x}$.
Match each of the following functions with the description of the quadrants in which their branches lie when plotted graphically: a) $y = \frac{5}{x}$ b)…
Match each of the following functions with the description of the quadrants in which their branches lie when plotted graphically:
- a) $y = \frac{5}{x}$
- b) $y = \frac{-2}{x}$
- c) $y = \frac{0,1}{x}$
Descriptions:
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Branches located in the first and third quadrants.
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Branches located in the second and fourth quadrants.
Look at the following graph of an inverse proportion function and determine its constant $k$ and its algebraic expression $y = \frac{k}{x}$.
Look at the following graph of an inverse proportion function and determine its constant $k$ and its algebraic expression $y = \frac{k}{x}$.
Translations: g(x) = k/(x-h) + v
A lottery prize of $300$ € is to be divided equally among a group of $x$ friends. Write the algebraic expression of the function that relates the money each person…
A lottery prize of $300$ € is to be divided equally among a group of $x$ friends. Write the algebraic expression of the function that relates the money each person receives to the number of friends, and calculate how much they receive if there are $6$ people.
To fill a swimming pool, $4$ taps with equal flow rate take $9$ hours. Write the function relating the filling time $t$ (in hours) to the number of open taps $n$. How…
To fill a swimming pool, $4$ taps with equal flow rate take $9$ hours. Write the function relating the filling time $t$ (in hours) to the number of open taps $n$. How long would $6$ taps take to fill the same pool?
According to Boyle's law, at constant temperature, the pressure $P$ of a gas and its volume $V$ are inversely proportional quantities ($P \cdot V = k$). If a given gas…
According to Boyle's law, at constant temperature, the pressure $P$ of a gas and its volume $V$ are inversely proportional quantities ($P \cdot V = k$). If a given gas occupies a volume of $20$ litres at a pressure of $1,5$ atmospheres, write the function $V(P)$ and calculate what volume it will occupy at $3$ atmospheres.
Monotonicity and graphing
The graph of the function $y = \frac{8}{x}$ is translated in the Cartesian plane $4$ units to the left and $3$ units downwards. Write the algebraic expression of the new…
The graph of the function $y = \frac{8}{x}$ is translated in the Cartesian plane $4$ units to the left and $3$ units downwards. Write the algebraic expression of the new function, determine its domain and find the coordinates of the point where its asymptotes intersect.
Find analytically the axes intercepts of the rational function $y = \frac{10}{x-2} + 5$.
Find analytically the axes intercepts of the rational function $y = \frac{10}{x-2} + 5$.
From the graph of this translated hyperbola, determine the equations of its asymptotes, the coordinates of its centre of symmetry and deduce its algebraic expression of…
From the graph of this translated hyperbola, determine the equations of its asymptotes, the coordinates of its centre of symmetry and deduce its algebraic expression of the form $y = \frac{k}{x-a} + b$.
Modelling with inverse proportionality
Inverse proportionality appears in inverse sharing (more workers, less time), the speed–time relationship over a fixed distance, or the pressure–volume relationship of a gas at constant temperature. The key step is to identify which product of the two quantities stays constant.
Express the rational function $y = \frac{2x+7}{x+3}$ in the form $y = \frac{k}{x-a} + b$ by performing polynomial division. Then, state the equations of its asymptotes…
Express the rational function $y = \frac{2x+7}{x+3}$ in the form $y = \frac{k}{x-a} + b$ by performing polynomial division. Then, state the equations of its asymptotes and the coordinates of its centre of symmetry.
Find analytically the points of intersection between the hyperbola $y = \frac{-4}{x-1}$ and the line $y = -x + 4$.
Find analytically the points of intersection between the hyperbola $y = \frac{-4}{x-1}$ and the line $y = -x + 4$.
An electronic components factory has daily fixed overheads of $500$ € to turn on the machinery, and producing each unit costs $2$ €. Write the average cost function per…
An electronic components factory has daily fixed overheads of $500$ € to turn on the machinery, and producing each unit costs $2$ €. Write the average cost function per unit $C_m(x)$ in terms of the number of units $x$ manufactured per day. Find its asymptotes and interpret the meaning of the horizontal asymptote in the context of the problem.