Inverse proportion function
If a rectangle always has the same area, the longer the base is, the shorter the height is: when you double one, the other is halved. This "opposite" relationship is inverse proportion, and its graph looks very different from lines and parabolas: two separate branches that never touch the axes.
The inverse proportion function
We work with $k\ne0$ and $x\ne0$. If $k=0$, then $0/x=0$ for $x\ne0$: there is no longer a hyperbola and its range is $\{0\}$. The controls below separate positive and negative values so that we can study the non-degenerate family.
Explore the positive control first and then the negative one.
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If $k > 0$: the branches lie in the first and third quadrants.
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If $k < 0$: the branches lie in the second and fourth quadrants.
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The greater $|k|$ is, the further the curve is from the axes.
Given the following table relating the base $x$ and the height $y$ of different rectangles with the same area: $x$: $2$, $3$, $4$, $6$ $y$: $12$, $8$, $6$, $4$ Determine…
Given the following table relating the base $x$ and the height $y$ of different rectangles with the same area:
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$x$: $2$, $3$, $4$, $6$
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$y$: $12$, $8$, $6$, $4$
Determine whether the relationship between $x$ and $y$ corresponds to an inverse proportion function. If so, find the constant of proportionality $k$ and write the algebraic expression for the function.
Find the algebraic expression of the inverse proportion function of the form $y = \frac{k}{x}$, knowing that its graph passes through the point $P(-2, 5)$.
Find the algebraic expression of the inverse proportion function of the form $y = \frac{k}{x}$, knowing that its graph passes through the point $P(-2, 5)$.
The asymptotes of the hyperbola
When the hyperbola is translated, the centre of symmetry (where the asymptotes would cross) is no longer at the origin:
Given the function $g(x) = \frac{3}{x+4} - 2$, state the equations of its horizontal and vertical asymptotes, and the coordinates of the centre of symmetry of the…
Given the function $g(x) = \frac{3}{x+4} - 2$, state the equations of its horizontal and vertical asymptotes, and the coordinates of the centre of symmetry of the hyperbola.
Consider the inverse proportionality function $f(x) = \frac{-12}{x}$, whose graphical representation is shown below: Based on its expression and graph, answer the…
Consider the inverse proportionality function $f(x) = \frac{-12}{x}$, whose graphical representation is shown below:
Based on its expression and graph, answer the following questions:
- a) State its domain and range.
- b) Write the equations of its asymptotes.
- c) Determine whether the function is increasing or decreasing on its domain.
Domain and range
Since we cannot divide by $0$, the function $y=\dfrac{k}{x}$ is not defined at $x=0$: its domain is $\mathbb{R}-\{0\}$. It also never reaches the value $y=0$ (the quotient is never exactly zero): its range is $\mathbb{R}-\{0\}$.
A train travels a fixed distance of $120\text{ km}$. Write the function relating the time $t$ (in hours) it takes to complete the journey as a function of the average…
A train travels a fixed distance of $120\text{ km}$. Write the function relating the time $t$ (in hours) it takes to complete the journey as a function of the average speed $v$ (in $\text{km/h}$). What type of function is it? Identify which is the independent variable and which is the dependent variable.
Express the rational function $f(x) = \frac{3x-7}{x-2}$ in the canonical form $f(x) = \frac{k}{x-a} + b$ by carrying out the corresponding polynomial division. Then,…
Express the rational function $f(x) = \frac{3x-7}{x-2}$ in the canonical form $f(x) = \frac{k}{x-a} + b$ by carrying out the corresponding polynomial division. Then, state the equations of its asymptotes and its domain.
Inverse proportion in real-life problems
Inverse proportion appears in inverse sharing (more workers, less time), speed and time over a fixed distance, or the pressure and volume of a gas. The key is to identify which product stays constant.
Find algebraically the points of intersection between the hyperbola $y = \frac{4}{x}$ and the line $y = -x + 5$.
Find algebraically the points of intersection between the hyperbola $y = \frac{4}{x}$ and the line $y = -x + 5$.
Determine analytically the points of intersection between the rational function $f(x) = \frac{2x+3}{x-1}$ and the line $g(x) = x+1$.
Determine analytically the points of intersection between the rational function $f(x) = \frac{2x+3}{x-1}$ and the line $g(x) = x+1$.
A rectangular wooden frame is to be built with an interior area of $24\text{ cm}^2$. Write the function relating the perimeter $P$ of the frame in terms of the length of…
A rectangular wooden frame is to be built with an interior area of $24\text{ cm}^2$.
Write the function relating the perimeter $P$ of the frame in terms of the length of its base $x$. Using the algebraic expression, determine whether the function has an oblique asymptote and explain its geometric meaning.