Some quantities have what we call an inverse relationship. As one of the quantities increases, the other one decreases, and vice versa.
A perfect example of this is the relationship between speed and time.
Say an oil tank driver needs to drive $1000$1000 km to deliver a load. The faster the driver travels, the less time it will take to cover this distance.
In the table below, we have used the formula $\text{Time }=\frac{\text{Distance }}{\text{Speed }}$Time =Distance Speed to find the time it would take to drive $1000$1000 km at different average speeds.
| Speed (km/h) | $60$60 | $70$70 | $80$80 | $90$90 | $100$100 | $110$110 | $120$120 |
| Time (hours) | $16.7$16.7 | $14.3$14.3 | $12.5$12.5 | $11.1$11.1 | $10$10 | $9.1$9.1 | $8.3$8.3 |
Plotting these values, we get a curve called a hyperbola:
Notice that if the driver were to travel at an average speed of $40$40 km/h, it would take $25$25 hours (or just over a day!) to complete the journey.
Each point on the curve corresponds to a different combination of speed and time, and all the points on the curve satisfy the equation $T=\frac{1000}{S}$T=1000S or $TS=1000$TS=1000, where $T$T is the time taken and $S$S is the speed travelled.
We can see from both equations that as average speed increases, the time taken will decrease. This is what we call inverse variation.
Inverse Variation
Two quantities $x$x and $y$y that are inversely proportional have an equation of the form:
$y=\frac{k}{x}$y=kx or $xy=k$xy=k or $x=\frac{k}{y}$x=ky,
where $k$k can be any number other than $0$0.
The number $k$k is called the constant of proportionality. In the case of the speed and time taken by the oil tank driver, the constant of proportionality was $1000$1000.
The graph of an inverse relationship in the $xy$xy-plane is called a hyperbola.
Graphs of hyperbolas
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).
Three graphs of inverse relationships - notice they each come in two pieces.
Notice the following features:
- Each hyperbola has two parts and they are 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 symmetric about the lines $y=x$y=x and $y=-x$y=−x, and they have rotational symmetry about the origin.
The quadrants of the hyperbola
When we consider an inverse relationship of the form $y=\frac{k}{x}$y=kx, there are two possible cases:
|
When $k$k is positive |
When $k$k is negative |
|---|---|
| Example: $y=\frac{2}{x}$y=2x or $xy=2$xy=2 | Example: $y=-\frac{3}{x}$y=−3x or $xy=-3$xy=−3 |
| When $x>0$x>0, $y>0$y>0 ⇒ Exists in 1st Quadrant | When $x>0$x>0, $y<0$y<0 ⇒ Exists in 4th Quadrant |
| When $x<0$x<0, $y<0$y<0 ⇒ Exists in 3rd Quadrant | When $x<0$x<0, $y>0$y>0 ⇒ Exists in 2nd Quadrant |
| When $k>0$k>0, the hyperbola $y=\frac{k}{x}$y=kx exists in the 1st and 3rd quadrants. | When $k<0$k<0, the hyperbola $y=\frac{k}{x}$y=kx exists in the 2nd and 4th quadrants. |
We can see this in the graphs of various hyperbolas:
The hyperbolas $y=\frac{2}{x}$y=2x and $y=\frac{5}{x}$y=5x are drawn in the positive-positive (first) and negative-negative (third) quadrants, while the hyperbolas $y=-\frac{1}{x}$y=−1x and $y=-\frac{3}{x}$y=−3x are drawn in the negative-positive (second) and positive-negative (fourth) quadrants.
The equation $y=\frac{k}{x}$y=kx and the hyperbola
We've seen that an inverse relationship between $x$x and $y$y can be described in three ways:
(1) $y=\frac{k}{x}$y=kx (2) $xy=k$xy=k (3) $x=\frac{k}{y}$x=ky
- If $x=0$x=0, the first equation becomes $y=\frac{k}{0}$y=k0, which is undefined. This explains the vertical asymptote on the graph. If the speed of the oil tank driver were $0$0 km/h, the driver wouldn't get anywhere and we couldn't talk about time taken.
- Similarly, if $y=0$y=0, the third equation becomes $x=\frac{k}{0}$x=k0, which is also undefined. This explains the horizontal asymptote on the graph. Thinking again of the oil tank driver, time could not be $0$0 as it wouldn't be possible for the driver to cover the $1000$1000 km distance in $0$0 hours.
In an inverse relationship, neither $x$x nor $y$y can ever be $0$0.
Inverse variation - A relation of the form $y=\frac{k}{x}$y=kx, where $k$k can be any number other than $0$0. In this relationship, as $x$x increases $y$y decreases, and vice-versa. The equation can also be written in the form $xy=k$xy=k or $x=\frac{k}{y}$x=ky.
Constant of proportionality - The value of $k$k in an inverse relationship.
Hyperbola - The graph of an inverse relationship. Has both vertical and horizontal asymptotes.
Asymptote - A line that the curve approaches but does not reach.
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
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 3
The equation $y=-\frac{6}{x}$y=−6x represents an inverse relationship between $x$x and $y$y.
Complete the table of values:
$x$x $-5$−5 $-4$−4 $-3$−3 $-2$−2 $-1$−1 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Is $y=-\frac{6}{x}$y=−6x increasing or decreasing when $x<0$x<0?
Increasing
ADecreasing
BDescribe the rate of increase when $x<0$x<0.
As $x$x increases, $y$y increases at a faster and faster rate.
AAs $x$x increases, $y$y increases at a slower and slower rate.
BAs $x$x increases, $y$y increases at a constant rate.
C