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.
A Cartesian plane has an $x$x-axis and $y$y-axis ranging from $-10$−10 to $10$10. Each axis has major tick marks at $2$2-unit intervals and minor tick marks at $1$1-unit intervals. A graph of the hyperbola $y=\frac{2}{x}$y=2x is plotted with two symmetrical branches. One branch lies in the first quadrant curving downward and rightward. And the other branch lies in the third quadrant curving downward and leftward.
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