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 $$1000 km to deliver a load. The faster the driver travels, the less time it will take to cover this distance.
In the table below, we have used the formula $$Time =Distance Speed to find the time it would take to drive $$1000 km at different average speeds.
| Speed (km/h) | $$60 | $$70 | $$80 | $$90 | $$100 | $$110 | $$120 |
| Time (hours) | $$16.7 | $$14.3 | $$12.5 | $$11.1 | $$10 | $$9.1 | $$8.3 |
Plotting these values, we get a curve called a hyperbola:
Notice that if the driver were to travel at an average speed of $$40 km/h, it would take $$25 hours (or just over a day!) to complete the journey.
Each point on the curve corresponds to a different combination of speed and time, and all the points on the curve satisfy the equation $$T=1000S or $$TS=1000, where $$T is the time taken and $$S is the speed travelled.
We can see from both equations that as average speed increases, the time taken will decrease. This is what we call inverse variation.
Inverse Variation
Two quantities $$x and $$y that are inversely proportional have an equation of the form:
$$y=kx or $$xy=k or $$x=ky,
where $$k can be any number other than $$0.
The number $$k is called the constant of proportionality. In the case of the speed and time taken by the oil tank driver, the constant of proportionality was $$1000.
The graph of an inverse relationship in the $$xy-plane is called a hyperbola.
Graphs of hyperbolas
Let's see what inverse variation looks like in a table of values.
This table shows the relationship $$y=1x:
| $$x | $$−4 | $$−2 | $$−1 | $$−0.5 | $$−0.25 | $$0.25 | $$0.5 | $$1 | $$2 | $$4 |
|---|---|---|---|---|---|---|---|---|---|---|
| $$y | $$−0.25 | $$−0.5 | $$−1 | $$−2 | $$−4 | $$4 | $$2 | $$1 | $$0.5 | $$0.25 |
Notice that:
- If $$x is positive, $$y is positive
- If $$x is negative, $$y is negative
- As $$x gets further from zero (in either direction), $$y gets closer to zero
Here are some hyperbolas with equations of the form $$y=kx (or $$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 $$y=0 (the $$x-axis), and they also approach the line $$x=0 (the $$y-axis), but they never cross these lines. This is because the equation $$y=kx is not defined for $$x=0 or $$y=0. We call the lines $$x=0 and $$y=0 asymptotes.
- They are symmetric about the lines $$y=x and $$y=−x, and they have rotational symmetry about the origin.
The quadrants of the hyperbola
When we consider an inverse relationship of the form $$y=kx, there are two possible cases:
|
When $$k is positive |
When $$k is negative |
|---|---|
| Example: $$y=2x or $$xy=2 | Example: $$y=−3x or $$xy=−3 |
| When $$x>0, $$y>0 ⇒ Exists in 1st Quadrant | When $$x>0, $$y<0 ⇒ Exists in 4th Quadrant |
| When $$x<0, $$y<0 ⇒ Exists in 3rd Quadrant | When $$x<0, $$y>0 ⇒ Exists in 2nd Quadrant |
| When $$k>0, the hyperbola $$y=kx exists in the 1st and 3rd quadrants. | When $$k<0, the hyperbola $$y=kx exists in the 2nd and 4th quadrants. |
We can see this in the graphs of various hyperbolas:
The hyperbolas $$y=2x and $$y=5x are drawn in the positive-positive (first) and negative-negative (third) quadrants, while the hyperbolas $$y=−1x and $$y=−3x are drawn in the negative-positive (second) and positive-negative (fourth) quadrants.
The equation $$y=kx and the hyperbola
We've seen that an inverse relationship between $$x and $$y can be described in three ways:
(1) $$y=kx (2) $$xy=k (3) $$x=ky
- If $$x=0, the first equation becomes $$y=k0, which is undefined. This explains the vertical asymptote on the graph. If the speed of the oil tank driver were $$0 km/h, the driver wouldn't get anywhere and we couldn't talk about time taken.
- Similarly, if $$y=0, the third equation becomes $$x=k0, which is also undefined. This explains the horizontal asymptote on the graph. Thinking again of the oil tank driver, time could not be $$0 as it wouldn't be possible for the driver to cover the $$1000 km distance in $$0 hours.
In an inverse relationship, neither $$x nor $$y can ever be $$0.
Inverse variation - A relation of the form $$y=kx, where $$k can be any number other than $$0. In this relationship, as $$x increases $$y decreases, and vice-versa. The equation can also be written in the form $$xy=k or $$x=ky.
Constant of proportionality - The value of $$k in an inverse relationship.
Hyperbola - The graph of an inverse relationship. Has both vertical and horizontal asymptotes.
Asymptote - A line that the curve approaches but does not reach.
Practice questions
Question 1
Which of the following equations represent inverse variation between $$x and $$y?
Select all correct answers.
$$y=7x
A$$y=6x+8
B$$y=−9x
C$$y=8x2
D$$y=2x2−7x−4
E$$y=3−x
F
Question 2
Consider the graph of $$y=2x.
For positive values of $$x, as $$x increases $$y approaches what value?
$$0
A$$1
B$$−∞
C$$∞
DAs $$x takes small positive values approaching $$0, what value does $$y approach?
$$∞
A$$0
B$$−∞
C$$π
DWhat are the values that $$x and $$y cannot take?
$$x$$=$$
$$y$$=$$
The graph is symmetrical across two lines of symmetry. State the equations of these two lines.
$$y=,y=
Question 3
The equation $$y=−6x represents an inverse relationship between $$x and $$y.
Complete the table of values:
$$x $$−5 $$−4 $$−3 $$−2 $$−1 $$y $$ $$ $$ $$ $$ Is $$y=−6x increasing or decreasing when $$x<0?
Increasing
ADecreasing
BDescribe the rate of increase when $$x<0.
As $$x increases, $$y increases at a faster and faster rate.
AAs $$x increases, $$y increases at a slower and slower rate.
BAs $$x increases, $$y increases at a constant rate.
C