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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$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.

 

Summary

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.

  1. $y=\frac{7}{x}$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=2x27x4

    E

    $y=3-x$y=3x

    F

Question 2

Consider the graph of $y=\frac{2}{x}$y=2x.

Loading Graph...

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

    $0$0

    A

    $1$1

    B

    $-\infty$

    C

    $\infty$

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

    $\infty$

    A

    $0$0

    B

    $-\infty$

    C

    $\pi$π

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

    $x$x$=$=$\editable{}$

    $y$y$=$=$\editable{}$

  4. 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.

  1. Complete the table of values:

    $x$x $-5$5 $-4$4 $-3$3 $-2$2 $-1$1
    $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Is $y=-\frac{6}{x}$y=6x increasing or decreasing when $x<0$x<0?

    Increasing

    A

    Decreasing

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

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

    A

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

    B

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

    C

Outcomes

MS2-12-1

uses detailed algebraic and graphical techniques to critically evaluate and construct arguments in a range of familiar and unfamiliar contexts

MS2-12-6

solves problems by representing the relationships between changing quantities in algebraic and graphical forms

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