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2.06 Inverse variation and hyperbolas

Lesson

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.

 

Summary

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.

  1. $$y=7x

    A

    $$y=6x+8

    B

    $$y=9x

    C

    $$y=8x2

    D

    $$y=2x27x4

    E

    $$y=3x

    F

Question 2

Consider the graph of $$y=2x.

Loading Graph...

  1. For positive values of $$x, as $$x increases $$y approaches what value?

    $$0

    A

    $$1

    B

    $$

    C

    $$

    D
  2. As $$x takes small positive values approaching $$0, what value does $$y approach?

    $$

    A

    $$0

    B

    $$

    C

    $$π

    D
  3. What are the values that $$x and $$y cannot take?

    $$x$$=$$

    $$y$$=$$

  4. 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.

  1. Complete the table of values:

    $$x $$5 $$4 $$3 $$2 $$1
    $$y $$ $$ $$ $$ $$
  2. Is $$y=6x increasing or decreasing when $$x<0?

    Increasing

    A

    Decreasing

    B
  3. Describe the rate of increase when $$x<0.

    As $$x increases, $$y increases at a faster and faster rate.

    A

    As $$x increases, $$y increases at a slower and slower rate.

    B

    As $$x increases, $$y increases at a constant rate.

    C

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