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 $$S is the speed travelled.
We can see from both equations that as average speed increases, the time taken will decrease. This is what we call inverse variation.
Two quantities $$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 $$xy-plane is called a hyperbola.
Let's see what inverse variation looks like in a table of values.
This table shows the relationship $$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:
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:
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 $$y=−3x are drawn in the negative-positive (second) and positive-negative (fourth) quadrants.
We've seen that an inverse relationship between $$x and $$y can be described in three ways:
(1) $$y=kx (2) $$xy=k (3) $$x=ky
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 $$k in an inverse relationship.
Hyperbola - The graph of an inverse relationship. Has both vertical and horizontal asymptotes.
Asymptote - A line that the curve approaches but does not reach.
Which of the following equations represent inverse variation between $$x and $$y?
Select all correct answers.
$$y=7x
$$y=6x+8
$$y=−9x
$$y=8x2
$$y=2x2−7x−4
$$y=3−x
Consider the graph of $$y=2x.
For positive values of $$x, as $$x increases $$y approaches what value?
$$0
$$1
$$−∞
$$∞
As $$x takes small positive values approaching $$0, what value does $$y approach?
$$∞
$$0
$$−∞
$$π
What 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.
$$y=,y=
The equation $$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
Decreasing
Describe the rate of increase when $$x<0.
As $$x increases, $$y increases at a faster and faster rate.
As $$x increases, $$y increases at a slower and slower rate.
As $$x increases, $$y increases at a constant rate.