2.04 Hyperbolas and inverse variation
Inverse variation
Inverse proportion, or inverse variation, means that as one amount increases the other amount decreases.
Mathematically, we write this as:
| $y$y | $\propto$∝ | $\frac{1}{x}$1x |
For example, speed and travel time vary inversely because the faster you go, the shorter your travel time.
We express these inverse variation relationships generally in the form
$y=\frac{k}{x}$y=kx
where $k$k is the constant of proportionality and $x$x and $y$y are any variables
It is also possible to write the equation in the form:
$xy=k$xy=k or $x=\frac{k}{y}$x=ky
and it is clear that $k$k can be any number other than $0$0.
The hyperbola
The graph of an inverse relationship in the $xy$xy-plane is called a hyperbola. Let's see what inverse variation looks like in a table of values.
This table shows the relationship $y=\frac{1}{x}$y=1x:
| $x$x | $-4$−4 | $-2$−2 | $-1$−1 | $-0.5$−0.5 | $-0.25$−0.25 | $0.25$0.25 | $0.5$0.5 | $1$1 | $2$2 | $4$4 |
|---|---|---|---|---|---|---|---|---|---|---|
| $y$y | $-0.25$−0.25 | $-0.5$−0.5 | $-1$−1 | $-2$−2 | $-4$−4 | $4$4 | $2$2 | $1$1 | $0.5$0.5 | $0.25$0.25 |
Notice that:
- If $x$x is positive, $y$y is positive
- If $x$x is negative, $y$y is negative
- As $x$x gets further from zero (in either direction), $y$y gets closer to zero
Here are some hyperbolas with equations of the form $y=\frac{k}{x}$y=kx (or $xy=k$xy=k).
Notice the following features:
- Each hyperbola has two parts and they lie in opposite quadrants.
- They all approach the line $y=0$y=0 (the $x$x-axis), and they also approach the line $x=0$x=0 (the $y$y-axis), but they never cross these lines. This is because the equation $y=\frac{k}{x}$y=kx is not defined for $x=0$x=0 or $y=0$y=0. We call the lines $x=0$x=0 and $y=0$y=0 asymptotes.
- They are symmetrical about the lines $y=x$y=x and $y=-x$y=−x, and they have rotational symmetry about the origin.
- We can see that every $x$x and $y$y coordinate on a hyperbola multiply to give the value of $k$k in their specific equation, as is suggested by the general form $xy=k$xy=k. (This gives us an easy method of finding an equation of a hyperbola from a sketch.)
Practice Questions
Question 1
Which of the following equations represent inverse variation between $x$x and $y$y?
Select all correct answers.
$y=\frac{7}{x}$y=7x
A$y=6x+8$y=6x+8
B$y=-\frac{9}{x}$y=−9x
C$y=\frac{8}{x^2}$y=8x2
D$y=2x^2-7x-4$y=2x2−7x−4
E$y=3-x$y=3−x
F
Question 2
In the table of values below, $m$m is proportional to $\frac{1}{p}$1p.
| $p$p | $2$2 | $4$4 | $5$5 | $x$x |
|---|---|---|---|---|
| $m$m | $140$140 | $y$y | $56$56 | $40$40 |
Determine the constant of proportionality, $k$k.
Using the solution to part (a), find the unknowns in the table:
$x$x$=$=$\editable{}$
$y$y$=$=$\editable{}$
Question 3
Consider the graph of $y=\frac{2}{x}$y=2x.
For positive values of $x$x, as $x$x increases $y$y approaches what value?
$0$0
A$1$1
B$-\infty$−∞
C$\infty$∞
DAs $x$x takes small positive values approaching $0$0, what value does $y$y approach?
$\infty$∞
A$0$0
B$-\infty$−∞
C$\pi$π
DWhat are the values that $x$x and $y$y cannot take?
$x$x$=$=$\editable{}$
$y$y$=$=$\editable{}$
The graph is symmetrical across two lines of symmetry. State the equations of these two lines.
$y=\editable{},y=\editable{}$y=,y=
Question 4
Consider the hyperbola that has been graphed.
Fill in the gap to complete the statement.
Every point $\left(x,y\right)$(x,y) on the hyperbola is such that $xy$xy$=$=$\editable{}$.
Considering that the relationship between $x$x and $y$y can be expressed as $xy=6$xy=6, which of the following is true?
If $x$x increases, $y$y must increase.
AIf $x$x increases, $y$y must decrease.
BWhich of the following relationships can be modelled by a function of the form $xy=a$xy=a?
The relationship between the number of people working on a job and how long it will take to complete the job.
AThe relationship between the number of sales and the amount of revenue.
BThe relationship between height and weight.
C